More about the Fodor-type Reﬂection Principle
Saka´e Fuchino, Hiroshi Sakai,
Lajos Soukup and Toshimichi Usuba
Abstract
Continuing the research began in Fuchino, Juh´asz, Soukup, Szent-mikl´ossy and Usuba [9], we study further the Fodor-type Reﬂection Prin-ciple (FRP) and its consequences.We show that FRP is equivalent to the non-existence of almost es-sentially disjoint ladder system on any stationary subset of a regularuncountable cardinal consisting of ordinals of countable coﬁnality (The-orem 2.7). Using this characterization, we show that FRP is actuallyequivalent to many known “mathematical” reﬂection theorems over ZFC.For example, it is shown that FRP is equivalent to the statement:“For any locally countably compact topological space X, if all subspacesof X of cardinality ≤ ℵ1 are metrizable, then X itself is also metrizable”(Theorem 2.8). Another example of statements equivalent to FRP is:“For any graph G, if all subgraphs of G of cardinality ≤ ℵ1 have count-able coloring number then G itself also has countable coloring number”(Theorem 3.1).We construct models of ZFC separating FRP from Reﬂection Prin-ciple (RP) and Ordinal Reﬂection Principle (ORP) (Theorems 5.2, 5.4and 6.1).
Date: May 28, 2012 (07:55 JST)
2010 Mathematical Subject Classiﬁcation: 03E35, 03E65, 54D20, 54D45, 54E35, 05C63
Keywords: Axiom R, strongly compact cardinal, reﬂection principle, locally compact,meta-Lindel¨of, metrizable, inﬁnite graph, coloring number, collectionwise Hausdorﬀ
The ﬁrst author was supported by Grant-in-Aid for Scientiﬁc Research (C) No. 19540152and No. 21540150 of the Ministry of Education, Culture, Sports, Science and TechnologyJapan (MEXT). The second author is supported by Grant-in-Aid for Grant-in-Aid for YoungScientists (B) No. 23740076 of the Ministry of Education, Culture, Sports, Science and Tech-nology Japan (MEXT). The third author is supported by the Hungarian National Foundationfor Scientiﬁc Research grant no. 61600, 68262 and 83726.
1Introduction
Let λ be a cardinal of coﬁnality ≥ ω1. The Fodor-type Reﬂection Principle forλ (FRP(λ)) introduced in [9] is the following statement:
FRP(λ) : For any stationary E ⊆ Eλω and mapping g : E → [λ]≤ℵ0 there isI ∈ [λ]ℵ1 such that
(1.1)cf(I) = ω1;
(1.2)g(α) ⊆ I for all α ∈ I ∩ E;
(1.3)for any regressive f : E ∩ I → λ such that f(α) ∈ g(α) for allα ∈ E ∩ I, there is ξ∗ < λ such that f −1 ′′{ξ∗} is stationary insup(I).
Here, for cardinals κ < λ, Eλκ denotes the set {α < λ : cf(α) = κ}. Similarly,we shall also write Eλ<κ, Eλ>κ, etc. to denote {α < λ : cf(α) < κ}, {α < λ :cf(α) > κ}, etc., respectively.In [9] it is shown that FRP(λ) is inconsistent for any singular λ. Thus wehave to deﬁne the global version of the Fodor-type Reﬂection Principle as:
FRP: FRP(λ) holds for all regular λ > ℵ1.
Note that FRP(ℵ1) is just a consequence of the Fodor’s Lemma in ZFC.The following local but cumulative version of FRP is proved to be useful (seee.g. Theorem 2.7 below). For any cardinal λ, let FRP(< λ) be the followingprinciple:
FRP(< λ): FRP(κ) holds for all regular uncountable κ < λ.
A mapping g : S → [λ]ℵ0 for some cardinal λ and S ⊆ λ is said to beessentially disjoint if there is a mapping f : S → [λ]<ℵ0 such that g(α) \ f(α),α ∈ S are pairwise disjoint. g is said to be almost essentially disjoint if g ↾ Dis essentially disjoint for all D ∈ [dom(g)]<| dom(g) |.Clearly, if g ↾ D is essentially disjoint and D′ ⊆ D then g ↾ D′ is alsoessentially disjoint. Hence if dom(g) is a regular cardinal λ then g is almostessentially disjoint if and only if g ↾ β is essentially disjoint for all β < λ.For a regular cardinal λ, the following principle was ﬁrst considered by S.Shelah. The notation with “ADS−” was used in [9] in analogy to the principleADS in [2]:
ADS−(λ): There are a stationary set S ⊆ λ and g : S → [λ]ℵ0 such that
(1.4)g(α) ⊆ α and otp(g(α)) = ω for all α ∈ S;
(1.5)g is almost essentially disjoint.
In [9], it is proved that, in the deﬁnition of ADS−(λ), we may assume thatS ⊆ Eλω.A mapping g : S → [λ]ℵ0 for some cardinal λ and S ⊆ Eλω is said to be aladder system if g(α) is a coﬁnal subset of α of order-type ω for all α ∈ S. Notethat a ladder system g : S → [λ]ℵ0 is essentially disjoint if and only if there is aregressive function f : S → λ such that g(α)\f(α), α ∈ S are pairwise disjoint.(1.4) of the deﬁnition of ADS−(λ), we may drop the condition otp(g(α)) = ωor further demand that g be a ladder system (see Lemma 2.5).In Section 2, we show that FRP(< λ) for any uncountable cardinal λ isequivalent to the statement that ADS−(µ) does not hold for all regular µ < λ(Theorem 2.7). Using this characterization, we show that many known reﬂectiontheorems in terms of topology and inﬁnite graph theory are actually equivalentto FRP: see (A) ∼ (E’) in Theorems 2.8, 3.1 and 4.1.Further equivalentassertions in topology and boolean algebras are given in [7], [8] and [10].In [9], it is proved that FRP(κ) follows from RP([κ]ℵ0) for all regular un-countable κ where RP([κ]ℵ0) is the following principle:
RP([κ]ℵ0) : For any stationary S ⊆ [κ]ℵ0, there is an I ∈ [κ]ℵ1 such that
(1.6)ω1 ⊆ I;
(1.7)cf(I) = ω1;
(1.8)S ∩ [I]ℵ0 is stationary in [I]ℵ0.
The global version RP of this principle is deﬁned as the assertion that RP([κ]ℵ0)holds for all uncountable cardinal κ of uncountable coﬁnality.The principle obtained from RP([κ]ℵ0) by excluding the condition (1.7) iscalled sometimes simply ‘reﬂection principle’.Adopting the terminology ofForeman and Todorcevic [6] or K¨onig, Larson and Yoshinobu [15], we shallcall this principle the Weak Reﬂection Principle for [κ]ℵ0 and denote it withWRP([κ]ℵ0). Its global version WRP is also deﬁned similarly, that is, WRPholds if and only if WRP([κ]ℵ0) holds for all uncountable κ. Since WRP enjoysthe downward transfer property: κ < κ′ ∧ WRP([κ′]ℵ0) ⇒ WRP([κ]ℵ0), RPimplies WRP.FRP is preserved by c.c.c. generic extension (see Theorem 3.4 in [9]). SinceWRP([ℵ2]ℵ0) implies 2ℵ0 ≤ ℵ2 (Todorcevic, see [13]), we obtain a model of FRP+¬WRP([ℵ2]ℵ0) by starting from a model of FRP and then forcing 2ℵ0 ≥ ℵ3
by any c.c.c. partial ordering. Unfortunately, it seems that we cannot modifythis method to prove the consistency of FRP with ¬RP (or ¬WRP) under2ℵ0 ≤ ℵ2. In, Section 5, we give several models which separate FRP and WRPunder 2ℵ0 ≤ ℵ2.Let us call the following reﬂection principle on stationarity of subsets of Eκωthe Ordinal Reﬂection Principle for κ and abbreviate it as ORP(κ):
ORP(κ): For any stationary S ⊆ Eκω, there is I ∈ [κ]ℵ1 such that (1.6), (1.7)and
(1.8)’ S ∩ I is stationary in sup(I).
The global version ORP of this principle is the assertion that ORP(κ) holdsfor every cardinal κ > ω1 of coﬁnality ≥ ω1. FRP(λ) implies ORP(λ). Bythe transfer property ORP(cf(κ)) ⇒ ORP(κ), we also have the implicationFRP ⇒ ORP. Shelah [23] proved that WRP([κ]ℵ0) implies ORP(κ).In Section 6, we construct a model of ¬FRP(κ) + ORP for arbitrary regularκ assuming MA+(σ-closed).The reﬂection principles we discussed above can be put in a yet larger pic-ture: It is obvious that RP follows from Axiom R of Fleissner ([4]) and AxiomR follows from MA+(σ-closed) (Beaudoin [1]).The known relations between these principles with some results from [9]as well as from Sections 5 and 6 of the present paper are summarized in thefollowing diagram:
MA+(σ-closed)Axiom RRPWRPORP
FRP2ℵ0 ≤ ℵ2
Section 5
Section 6
MA+(σ-closed)does not imply ¬CH[20][17]
Recently we obtained a result which further extends the diagram above: In[11] it is namely proved that FRP follows from Rado Conjecture (RC). Theseparation of RC and FRP can be easily obtained: Since RC implies 2ℵ0 ≤ ℵ2([24]) any model of FRP and 2ℵ0 > ℵ2 would separate these principles. Under2ℵ0 = ℵ2 a model of MM would provide such a separation.The model inTheorem 5.4 gives the separation under CH since it is known that RC impliesRP([ℵ2]ℵ0) (see [24]).
2Further characterizations of FRP
For a set X, we call a sequence ⟨Xα : α < κ⟩ a ﬁltration of X if
(2.1)κ = cf(| X |);
(2.2)⟨Xα : α < κ⟩ is a continuously increasing sequence of subsets of X ofcardinality < | X |; and
(2.3)X = ∪
α<κ Xα.
For I ∈ [λ]ℵ1 with cf(I) = ω1, we can ﬁnd a sup-increasing ﬁltration ⟨Iξ : ξ <ω1⟩. Here, for a set S of ordinals, a sequence ⟨Sξ : ξ < κ⟩ of subsets of S forκ = cf(| S |) is said to be a sup-increasing ﬁltration if ⟨Sξ : ξ < κ⟩ is a ﬁltrationof S and
(2.4)⟨sup(Sξ) : ξ < κ⟩ is strictly increasing.
Lemma 2.1. Suppose that λ is a regular cardinal > ℵ1, E ⊆ Eλω is stationaryin λ and g : E → [λ]ℵ0 is such that
(2.5)g(α) ∩ α is coﬁnal in α for all α ∈ E.
(2.6)I ∈ [λ]ℵ1 is closed with respect to g and
(2.7)cf(I) = ω1,
then the following are equivalent:
(a) For any f : E ∩ I → I such that f(α) ∈ g(α) ∩ α for all α ∈ E ∩ I,there is ξ∗ ∈ I such that f −1 ′′{ξ∗} is stationary in sup(I);
(b) For any f : E ∩ I → I such that f(α) ∈ g(α) ∩ α for all α ∈ E ∩ I,there is ξ∗ ∈ I such that f −1 ′′{ξ∗} is coﬁnal in sup(I);
(c) The set Z = {x ∈ [I]ℵ0 : sup(x) ∈ E ∩ I and g(sup(x)) ∩ sup(x) ⊆ x}is stationary in [I]ℵ0.
Note that (a) above is identical with the condition (1.3) in the deﬁnition ofFRP(λ).
Proof. “(a) ⇒ (b)” is clear. So we shall prove “(c) ⇒ (a)” and “(b) ⇒ (c)”.“(c) ⇒ (a)”: Assume that I satisﬁes (c) and suppose that f : E ∩ I → λ issuch that f(α) ∈ g(α) ∩ α for all α ∈ E ∩ I. Since
Z′ = {x ∈ [I]ℵ0 : sup(x) ∈ E ∩ I and f(sup(x)) ∈ x} ⊇ Z,
Z′ is stationary in [I]ℵ0 by the assumption. By Fodor’s Lemma, it follows thatthere is a ξ∗ ∈ I such that
Z′′ = {x ∈ Z′ : sup(x) ∈ E ∩ I and f(sup(x)) = ξ∗}
is stationary in [I]ℵ0. Then f −1 ′′{ξ∗} = {α ∈ sup(I) : sup(x) = α for some x ∈Z′′} is stationary in sup(I). Note that we need here the condition (2.7).
“(b) ⇒ (c)”: We prove the contraposition. Assume that (c) does not hold.We show that (b) does not hold then. Since (c) does not hold, there is a sup-increasing ﬁltration ⟨Iη : η < ω1⟩ of I such that, for all η < ω1,
(2.8)if sup(Iη) ∈ E ∩ I then g(sup(Iη)) ∩ sup(Iη) ̸⊆ Iη.
Let f : E ∩ I → λ be deﬁned by
(2.9)f(α) = min ((g(α) ∩ α) \ Iηα),
where ηα = sup{η < ω1 : sup(Iη) < α} for α ∈ E ∩ I. Note that f is well-deﬁned since, for α ∈ E ∩ I, if sup(Iηα) < α then (g(α) ∩ α) \ Iηα ̸= ∅ by (2.5);otherwise, we have again (g(α) ∩ α) \ Iηα ̸= ∅ by (2.8). By the deﬁnition (2.9),f(α) ∈ g(α) ∩ α for all α ∈ E ∩ I. So the following claim shows that this f is acounterexample to (b):
Claim 2.1.1. f −1 ′′{ξ} is bounded in sup(I) for all ξ ∈ I.
⊢For ξ ∈ I, let η∗ = min{η < ω1 : ξ ∈ Iη}. Then we have ξ ∈ Iη∗ ⊆ Iηαfor all α ∈ (E ∩ I) \ sup(Iη∗). By (2.9), it follows that f(α) ̸= ξ for all α ∈(E ∩ I) \ sup(Iη∗). Thus we have f −1 ′′{ξ} ⊆ sup(Iη∗).⊣ (Claim 2.1.1)
(Lemma 2.1)
The following proposition gives one of the seemingly weakest assertionsamong the diverse reformulations of FRP(λ).
Proposition 2.2. For a regular λ > ℵ1, FRP(λ) is equivalent to the followingassertion:
(2.10)λ For any stationary E ⊆ Eλω and a ladder system g : E → [λ]ℵ0, thereis an I ∈ [λ]ℵ1 such that
(2.10a) cf(I) = ω1;
(2.10b) I is closed with respect to g;
(2.10c) for any f : E ∩I → I such that f(α) ∈ g(α) for all α ∈ E ∩I,there is a ξ∗ ∈ I such that f −1 ′′{ξ∗} is coﬁnal in sup(I).
Proof. It is clear that FRP(λ) implies (2.10)λ. To show the other implication,assume (2.10)λ and suppose that E ⊆ Eλω is stationary and g : E → [λ]ℵ0. Wehave to show that there is an I as in the deﬁnition of FRP(λ) for these E andg.Without loss of generality, we may assume that g(α) ∩ α is coﬁnal in α forevery α ∈ E.Let h : λ → λ be a λ to 1 surjection and let
(2.11) C = {α < λ : (a)α is closed with respect to g and h;(b){γ < α : h(γ) = β} is coﬁnal in α for all β < α }.
C is a club subset of λ: To see that C is unbounded in λ, note that, for asuﬃciently large regular θ and M ≺ H(θ) with g, h ∈ M and λ ∩ M ∈ λ, wehave λ ∩ M ∈ C.It follows that E0 = E ∩ C is stationary. For α ∈ E0, let {ξαi : i ∈ ω} be anenumeration of g(α) ∩ α and let ⟨ηαi : i ∈ ω⟩ be a strictly increasing sequenceof ordinals coﬁnal in α such that h(ηαi ) = ξαi . This is possible since α ∈ C andthus α satisﬁes (2.11), (b). Let g0 : E0 → [λ]ℵ0 be the ladder system deﬁned by
(2.12) g0(α) = {ηαi : i ∈ ω} for α ∈ E0.
By the assumption (2.10)λ there is an I0 ∈ [λ]ℵ1 satisfying (2.10)λ (forE = E0 and g = g0). Let I be the closure of I0 with respect to g. Since I0satisﬁes (2.10c) for E = E0 and g = g0, E0 ∩ I0 is coﬁnal in I0. It follows thatsup(I0) ∈ C and, by (2.11), (a), we have sup(I) = sup(I0).We claim that this I satisﬁes (1.3). Suppose that f : E ∩ I → λ is such thatf(α) ∈ g(α) ∩ α for all α ∈ E ∩ I. Let f0 : I0 ∩ E0 → λ be deﬁned by
(2.13) f0(α) ∈ g0(α) and h(f0(α)) = f(α)
for all α ∈ E0 ∩ I0. In particular, we have then
(2.14) h ◦ f0 = f ↾ E0 ∩ I0.
Since I0 was chosen according to (2.10)λ, there is ξ∗ ∈ I0 such that f −10′′{ξ∗} iscoﬁnal in sup(I0) = sup(I). By (2.14), it follows that f −1 ′′{h(ξ∗)} is coﬁnal insup(I). Thus we have shown that I satisﬁes (b) of Lemma 2.1. By Lemma 2.1,it follows that I satisﬁes (1.3).(Proposition 2.2)
Lemma 2.3. For a regular cardinal λ > ℵ1, a stationary E ⊆ Eλω, a mappingg : E → [λ]ℵ0 such that g(α) is a coﬁnal subset of α for all α ∈ E, and α∗ ∈ Eκω1,the following are equivalent:
(a) There is I ∈ [α∗]ℵ1 such that sup(I) = α∗, I is closed with respect to gandZI = {x ∈ [I]ℵ0 : sup(x) ∈ E and g(sup(x)) ⊆ x}
is stationary;
(a’) For any I ∈ [α∗]ℵ1 such that sup(I) = α∗ and I is closed with respectto g as well as with respect to the order topology of α∗, we have that
ZI = {x ∈ [I]ℵ0 : sup(x) ∈ E and g(sup(x)) ⊆ x}
is stationary;
(b) The set
Zα∗ = {x ∈ [α∗]ℵ0 : sup(x) ∈ E and g(sup(x)) ⊆ x}
is stationary.
Proof. “(a’) ⇒ (a)” is clear.“(a) ⇒ (b)”: Suppose that Zα∗ is not stationary and let C ⊆ [α∗]ℵ0 be aclub disjoint from Zα∗. Let I ∈ [α∗]ℵ1 be such that I is coﬁnal in α∗ and closedwith respect to g. Let
(2.15) C′ = {x ∩ I : x ∈ C and sup(x) = sup(x ∩ I)}.
Then we can ﬁnd a C′′ ⊆ C′ which is a club in [I]ℵ0. C′′ is still disjoint fromZα∗ and hence also from ZI. Thus ZI is not stationary.
“(b) ⇒ (a’)”: Assume that Zα∗ is stationary. Let I ∈ [α∗]ℵ1 be such thatsup(I) = α∗ and I is closed with respect to g as well as with respect to theorder topology of α∗. We have to show that ZI is stationary in [I]ℵ0. Supposethat C ⊆ [I]ℵ0 is a club. Let
(2.16)˜C = {x ∪ y : x ∈ C, y ∈ [α∗ \ I]ℵ0, sup(x) ≥ sup(y)}.
Then ˜C is a club in [α∗]ℵ0. Hence, by the assumption, there is z ∈ Zα∗ ∩ ˜C. Letx = z ∩ I. By (2.16) and since I is closed with respect to the order topologyof α∗, we have sup(z) = sup(x) ∈ E ∩ I. By closedness of I with respect tog, it follows that g(sup(x)) ⊆ I. Hence g(sup(x)) ⊆ z ∩ I = x. Thus we havex ∈ ZI ∩ C. This shows that ZI is stationary.(Lemma 2.3)
Proposition 2.4. For a regular λ > ℵ1, FRP(λ) is equivalent to the assertion:
(2.17)λ For any stationary E ⊆ Eλω and a ladder system g : E → [λ]ℵ0, thereis α∗ ∈ Eλω1 such that
{x ∈ [α∗]ℵ0 : sup(x) ∈ E, g(sup(x)) ⊆ x}
is stationary in [α∗]ℵ0.
Proof. By Proposition 2.2, FRP(λ) is equivalent to (2.10)λ. The equivalence of(2.17)λ to (2.10)λ follows from Lemma 2.3 and Lemma 2.1.(Proposition 2.4)
Lemma 2.5. Suppose that λ is a regular cardinal. Then ADS−(λ) is equivalentto the following assertion:
(2.18)λ There are a stationary E∗ ⊆ Eλω and a ladder system g∗ : E∗ → [λ]ℵ0
such that
(2.18a) g∗ is almost essentially disjoint, i.e., g∗ ↾ α is essentially dis-joint for all α < λ.
Proof. “(2.18)λ ⇒ ADS−(λ)” is trivial. The proof of “ADS−(λ) ⇒ (2.18)λ”can be done similarly to the proof of Proposition 2.2 using the λ to 1 surjectionargument.(Lemma 2.5)
Proposition 2.6. Suppose that FRP does not hold and let
λ∗ = min{µ : µ is regular and FRP(µ) does not hold}.
Then we have ADS−(λ∗).
In [9], it is proved that, for all regular µ, FRP(µ) implies ¬ADS−(µ). ThusProposition 2.6 implies the following characterization of FRP:
Theorem 2.7. FRP(< λ) is equivalent to the assertion:
(2.19) ADS−(µ) does not hold for all regular µ < λ.
Proof of Proposition 2.6: First note that λ∗ is regular and ≥ ℵ2. By Propo-sition 2.2 and Lemma 2.1, there are a stationary E ⊆ Eλ∗ω and a ladder systemg : E → [λ∗]ℵ0 such that, for any I ∈ [λ∗]ℵ1 closed with respect to g withcf(I) = ω1, we have
(2.20) ZI = {x ∈ [I]ℵ0 : sup(x) ∈ E ∩ I and g(sup(x)) ⊆ x} is non-stationaryin [I]ℵ0.
Let σ : λ∗ → ℵ0>λ∗ be a λ∗ to 1 surjection and
(2.21) C∗ = {α < λ∗ : for all a ∈ ℵ0>α, {γ < α : σ(γ) = a} is coﬁnal in α}.
Similarly to the argument after (2.11), we can show that C∗ is a club subset ofλ∗. Thus E∗ = E ∩ C∗ is a stationary subset of λ∗.For α ∈ E∗, let ⟨ηαi : i < ω⟩ be an increasing sequence of ordinals coﬁnal inα such that
(2.22) σ(ηαi ) = ⟨ξαk : k ≤ i⟩
where ⟨ξαk : k < ω⟩ is a ﬁxed enumeration of g(α). Let g∗ : E∗ → [λ∗]ℵ0 be theladder system deﬁned by
(2.23) g∗(α) = {ηαi : i < ω}.
We show that g∗ is almost essentially disjoint. To this end, we prove a series ofclaims.
Claim 2.6.1. Suppose that I ∈ [λ∗]ℵ1, cf(I) = ω1 and I is closed with respect tog and g∗. For a suﬃciently large regular θ, let N ≺ H(θ) be such that | N | = ℵ0and I, σ, λ∗, g, g∗ ∈ N. Then we have | g∗(α)∩N | < ℵ0 for all α ∈ (E∗∩I)\N.
⊢We prove the claim by induction on sup(I).Since g∗(α) is a coﬁnal subset of α ∩ I of order-type ω, the claim is trivial ifα > sup(α ∩ N). So let us assume
(2.24) α = sup(α ∩ N).
Case I: α = sup(I ∩ N).
By (2.20) and by elementarity of N, there is a sup-increasing ﬁltration ⟨Iξ :ξ < ω1⟩ ∈ N of I such that
(2.25) for all ξ < ω1, if sup(Iξ) ∈ E then g(sup(Iξ)) ̸⊆ Iξ.
For ζ = N ∩ ω1 we have I ∩ N = Iζ and α = sup(Iζ). By (2.25), there is k0 ∈ ωsuch that ξαk0 ̸∈ Iζ. Then we have ηαi ̸∈ N for all i ∈ ω \ k0 by (2.22) and henceby (2.23) | g∗(α) ∩ N | < ℵ0.
Case II: α < sup(I ∩ N).
Let δ = min((I ∩ N) \ α). Then I ∩ δ ∈ N and α ∈ (I ∩ δ) \ sup((I ∩ δ) ∩ N).Hence cf(I ∩ δ) = ω1. Since sup(I ∩ δ) < sup(I), we may apply the inductionhypothesis to I ∩ δ to conclude | g∗(α) ∩ N | < ℵ0.⊣ (Claim 2.6.1)
Claim 2.6.2. If I ∈ [λ∗]ℵ1 is such that cf(I) = ω1 and I is closed with respectto g and g∗, then there is a sup-increasing ﬁltration ⟨Iξ : ξ < ω1⟩ of I suchthat, for all ξ < ω1,
(2.26) Iξ is closed with respect to g and g∗;
(2.27) | g∗(α) ∩ Iξ | < ℵ0 for every α ∈ (E∗ ∩ I) \ Iξ.
⊢Let θ be a suﬃciently large regular cardinal and let ⟨Nξ : ξ < ω1⟩ be acontinuously increasing chain of countable elementary submodels of H(θ) suchthat I, σ, λ∗, g, g∗ ∈ N0 and I ⊆ ∪
ξ<ω1 Nξ.Then, by Claim 2.6.1, Iξ = I ∩ Nξ for all ξ < ω1 are as desired.⊣ (Claim 2.6.2)
Claim 2.6.3. For any I ∈ [λ∗]ℵ1, g∗ ↾ (E∗ ∩ I) is essentially disjoint.
⊢By blowing up I if necessary, we may assume that cf(I) = ω1 and I is closedwith respect to g and g∗. Let ⟨Iξ : ξ < ω1⟩ be a ﬁltration of I as in Claim2.6.2. For ξ < ω1, let ⟨ρξn : n < ω⟩ be an enumeration of E∗ ∩ (Iξ+1 \ Iξ) wherewe assume without loss of generality that | E∗ ∩ (Iξ+1 \ Iξ) | = ℵ0 for all ξ < ω1.By (2.27), for each n < ω there is ζξn < ρξn such that (g∗(ρξn) \ ζξn) ∩ Iξ = ∅.We can ﬁnd νξn with ζξn ≤ νξn < ρξn such that g∗(ρξn) \ νξn is disjoint from g∗(ρξi)for all i < n.Now, for α ∈ E∗ ∩ I, let ξ < ω1 and n < ω be such that α = ρξn and letf(α) = νξn. Then g∗(α)\f(α), α ∈ E∗∩I are pairwise disjoint. ⊣ (Claim 2.6.3)
Claim 2.6.4. Suppose δ < λ∗, κ = cf(δ) ≥ ℵ1 and W ∈ [δ]κ is such that W isclub in δ and closed with respect to g and g∗. If ⟨Wξ : ξ < κ⟩ is a sup-increasingﬁltration of W then
C = {ξ < κ : for all α ∈ E∗ ∩ (W \ Wξ), | g∗(α) ∩ Wξ | < ℵ0}
contains a club subset of κ.
⊢Assume, toward a contradiction, that κ \ C is stationary.Let S = (κ \ C) ∩ Eκω.
Subclaim 2.6.4.1. S is stationary.
⊢If C is bounded then this is clear.Otherwise, let D = {ξ < κ : C ∩ξ is unbounded in ξ}. Then D is a club subset of κ and D ∩ Eκ>ω ⊆ C. Sinceκ \ C is assumed to be stationary, it follows that S must be stationary.⊣ (Subclaim 2.6.4.1)
Note that, for all ξ ∈ S, there is αξ ∈ W \ Wξ such that
(2.28) | g∗(αξ) ∩ Wξ | = ℵ0.
Without loss of generality, we may assume that αξ ∈ Wξ+1 \ Wξ.Let π : κ → W be a bijection such that π−1 ′′Wξ ∈ κ for all ξ < κ. Letπ−1 ′′Wξ = βξ for ξ < κ and B = {βξ : ξ ∈ S}. Then B is a stationary subsetof Eκω.For β ∈ B with β = βξ for some ξ < κ, let g′(β) = π−1 ′′(g∗(αξ)∩Wξ). Then,by Claim 2.6.3, g′ ↾ (B ∩I) is essentially disjoint for any I ∈ [κ]ℵ1. In particularB and g′ make up a counterexample to FRP(κ). But this is a contradictionsince we have FRP(κ) by cf(κ) = κ < λ∗.⊣ (Claim 2.6.4)
Now, we can show that g∗ ↾ (E∗ ∩ δ) is essentially disjoint for all δ < λ∗ byinduction on δ.For δ = 0, this is trivial. If the essential disjointness holds for an ordinalδ ∈ λ∗ \ E∗, then we have E∗ ∩ δ = E∗ ∩ (δ + 1) and hence the essentialdisjointness also holds for δ + 1.If δ ∈ E∗ then let f : E∗ ∩ δ → λ∗ be a regressive function witnessingthe essential disjointness for δ. For each α ∈ E∗ ∩ δ, let να < α be such thatg∗(α) \ να and g∗(δ) are disjoint and let f ∗ : E∗ ∩ (δ + 1) → λ∗ be deﬁned by
f ∗(α) =
{max{f(α), να};if α ∈ δ,
0;otherwise (i.e. if α = δ).
Then f ∗ witnesses the essential disjointness for δ + 1.Suppose now that δ is a limit ordinal and we have shown the essentialdisjointness for all δ′ < δ.
Case I. cf(δ) = ω.
Let ⟨δn : n < ω⟩ be a strictly increasing sequence of ordinals coﬁnal in δ. Foreach n ∈ ω, let fn : E∗ ∩ (δn + 1) → δn be a regressive function witnessingthe essential disjointness of g∗ ↾ (E∗ ∩ (δn + 1)). Let f ∗0 = f0 and let f ∗n+1 :E∗ ∩ ((δn+1 + 1) \ (δn + 1)) → λ∗ be deﬁned by f ∗n+1(α) = max{fn+1(α), δn}for α ∈ E∗ ∩ ((δn+1 + 1) \ (δn + 1)). Then it is easy to see that f = ∪
n∈ω f ∗nwitnesses the essential disjointness for δ.
Case II. cf(δ) > ω.
Let κ = cf(δ) and let W be a club subset of δ of cardinality κ which is closedwith respect to g and g∗. By Claim 2.6.4, there is a sup-increasing ﬁltration⟨Wξ : ξ < κ⟩ of W such that, for every ξ < κ and α ∈ E∗ ∩ (W \ Wξ), we have| g∗(α) ∩ Wξ | < ℵ0. For ξ < κ, let ηξ = sup(Wξ).
For α ∈ E∗ ∩ W, let ξα < κ be such that α ∈ Wξα+1 \ Wξα and let να < αbe such that (g∗(α) \ να) ∩ Wξα = ∅. For α ∈ (E∗ ∩ δ) \ W, let ξ∗α < κ be suchthat ηξ∗α ≤ α < ηξ∗α+1. Note that actually we have ηξ∗α < α in this case: Sinceα ̸∈ W and W is closed, α is not a limit of W. Thus there is a µα < α suchthat (g∗(α) \ µα) ∩ W = ∅.By induction hypothesis, there is a regressive fξ : E∗ ∩ ηξ → ηξ witnessingthe essential disjointness of g∗ ↾ (E∗∩ηξ) for every ξ < κ. Now let f : E∗∩δ → δbe deﬁned by
(2.29) f(α) =
{max{fξα+1(α), να};if α ∈ W,
max{fξ∗α+1(α), ηξ∗α, µα};otherwise
for α ∈ E∗ ∩ δ. Then f witnesses the essential disjointness of g∗ ↾ E∗ ∩ δ.
(Proposition 2.6)
In virtue of FRP in Theorem 2.7 we can also obtain many “mathematical”characterizations of FRP.For the notions and notations used in (A) and (B) see [9]; for (C) and (C’),see Fleissner [4] or Fuchino [7].
Theorem 2.8. For a cardinal λ, FRP(< λ) is equivalent to each of the followingassertions over ZFC:
(A) For every locally separable countably tight topological space X with L(X) <λ, if all subspaces of X of cardinality ≤ ℵ1 are meta-Lindel¨of, then Xitself is also meta-Lindel¨of.
(B) For every locally countably compact topological space X with L(X) < λ,if all subspaces of X of cardinality ≤ ℵ1 are metrizable, then X itself isalso metrizable.
(C) For every T1-space X of cardinality < λ with point countable base, ifall subspaces of X of cardinality ≤ ℵ1 are left-separated then X itself isalso left-separated.
(C’) For every metrizable space X of cardinality < λ, if all subspaces of Xof cardinality ≤ ℵ1 are left-separated then X itself is also left-separated.
Proof. The implications “FRP(< λ) ⇒ (A), (B)” are proved in Fuchino et al.[9]. It is also proved in [9] that ADS−(λ∗) implies the existence of a topologicalspace X of cardinality λ∗ which is a counter-example to both of (A) and (B).Note that we have L(X) ≤ | X | = λ∗. Hence, it follows from Theorem 2.7 thatboth of the assertions (A) and (B) are equivalent to FRP(< λ).
The implication “FRP(< λ) ⇒ (C)” is proved in Fuchino [7]. “(C) ⇒ (C’)”is trivial noting that every metric spaces are T1 and have a point countablebase by Stone’s theorem. For the implication “(C’) ⇒ FRP(< λ)”, assume thatADS−(λ∗) holds for a regular cardinal λ∗ and we show that there is a topologicalspace X of cardinality λ∗ which is a counter-example to the assertion of (C’).Let E∗ and g∗ be such that E∗ is a stationary subset of Eλ∗ω and g an almostessentially disjoint ladder system on E∗.The topological space given below is a modiﬁcation of an example given inFleissner [4].Let X = E∗ be the metric space with the metric d deﬁned by
d(α, β) =1
kα,β + 1
for α, β ∈ X where
kα,β = min{k < ω : the set of the ﬁrst k elements of g∗(α) and the setof the ﬁrst k elements of g∗(β) are distinct}.
Let O be the topology of X which is induced from d. We show that (X, O) isa counter-example to the assertion (C’).Since λ∗ is a regular cardinal and subspaces of a left-separated space arealso left-separated, the next claim implies that all subspaces of X of cardinality< λ∗ are left-separated:
Claim 2.8.1. E∗ ∩ β as a subspace of X is left-separated for all β < λ∗.
⊢Let β < λ∗. Since g∗ is almost essentially disjoint, there is a regressivef : E∗ ∩ β → β such that g∗(α) \ f(α), α ∈ E∗ ∩ β are pairwise disjoint.For n ∈ ω, let
Dn = {α ∈ E∗ ∩ β : g∗(α) ∩ f(α) consists ofthe ﬁrst n elements of g∗(α)}.
Then E∗ ∩ β = ∪
n∈ω Dn. Each Dn is discrete since d(α, α′) ≥1
n+1 for every α,α′ ∈ Dn. By Theorem 2.2 in [4], it follows that E∗ ∩ β is left-separated.⊣ (Claim 2.8.1)
To show that (X, O) itself is not left-separated, it is enough to show thefollowing claim. Note that there are stationarily many α ∈ E∗ which can berepresented as αM as below. Also note that it is stated implicitly in Theorem2.2. in [4] that a T1-space X with point countable base is left-separated if andonly if it is left-separated in order-type | X |.
Claim 2.8.2. Let θ be a suﬃciently large regular cardinal. If M ≺ H(θ) issuch that | M | < λ∗, λ∗, E∗, g ∈ M and αM = λ∗ ∩ M ∈ E∗. Then we haveαM ∈ αM ∩ E∗.
⊢Let ⟨αn : n ∈ ω⟩ be an increasing enumeration of g∗(αM). For each k ∈ ω,let ψk be the formula asserting
“there exists an α ∈ E∗ such that the ﬁrst k elements of g∗(α) areα0,. . . , αk−1”.
H(θ) |= ψk holds for each k ∈ ω since α∗ is a witness of all of ψk, k ∈ ω. Itfollows that M |= ψk for each k ∈ ω by elementarity. But, by the deﬁnitionof d, this means that there is α ∈ αM ∩ E∗ such that d(αM, α) ≤1
k+1 for eachk ∈ ω.⊣ (Claim 2.8.2)
(Theorem 2.8)
3Coloring number of graphs
We consider here a graph G as a pair ⟨G, E⟩ where the elements of G are pointsand the elements of E ⊆ [G]2 represent the edges of the graph. For a graphG = ⟨G, E⟩, the coloring number of G is deﬁned by:
(3.1)col(G) = min{µ : there is a well-ordering ≺ of G such that| {y ∈ G : y ≺ x and {x, y} ∈ E} | < µ for all x ∈ G}.
For a graph G = ⟨G, E⟩ and I ⊆ G, we denote with G ↾ I the subgraph⟨G ∩ I, E ∩ [I]2⟩ of G. Sometimes we also identify a subset G′ of G with thesubgraph ⟨G′, E ∩ [G′]2⟩. For I ⊆ G and x ∈ G we write:
ExI = {y ∈ I : {x, y} ∈ E}.
The following is the main theorem of this section:
Theorem 3.1. For any cardinal λ ≥ ℵ2, FRP(< λ) is equivalent to the asser-tion:
(D) For any graph G = ⟨G, E⟩ of cardinality < λ, if col(G ↾ I) ≤ ℵ0 for allI ∈ [G]≤ℵ1, then col(G) ≤ ℵ0.
Let us begin with the characterization of coloring number ≤ µ by Erd˝os andHajnal (Lemma 3.4).
Lemma 3.2. Suppose that G = ⟨G, E⟩ is a graph and f : G → [G]<µ for acardinal µ is such that
(3.2)at least one of x ∈ f(y) and y ∈ f(x) holds for all {x, y} ∈ E.
If G′ ⊆ G is closed with respect to f then | ExG′ | < µ for all x ∈ G \ G′.
Proof. Suppose that x ∈ G \ G′. If x′ ∈ G′ then f(x′) ⊆ G′ by the closednessof G′ with respect to f. Thus we have x ̸∈ f(x′). Hence, if {x, x′} ∈ E, then wemust have x′ ∈ f(x). It follows that ExG′ ⊆ f(x) and | ExG′ | ≤ | f(x) | < µ.
(Lemma 3.2)
Lemma 3.3. Suppose that G = ⟨G, E⟩ is a graph of cardinality λ, κ = cf(λ) andµ an inﬁnite cardinal. If ⟨Gα : α < κ⟩ is a ﬁltration of G such that col(Gα) ≤ µfor all α < κ and | ExGα | < µ for all α < κ and x ∈ Gα+1 \ Gα, then we havecol(G) ≤ µ and G has a well-ordering ≺ of order-type λ witnessing col(G) ≤ µ.
Proof. For each α < κ, let ≺α be a well-ordering of Gα+1 witnessing col(Gα+1) ≤µ. That is, ≺α is such that
| {y ∈ Gα+1 : y ≺α x and {x, y} ∈ E} | < µ for all x ∈ Gα+1.
Let ≺ be the well-ordering of G deﬁned by
x ≺ y ⇔ (x, y ∈ Gα+1 \ Gα and x ≺α y for some α < κ) or(x ∈ Gα and y ̸∈ Gα for some α < κ)
for x, y ∈ G. Then ≺ witnesses col(G) ≤ µ:
Claim 3.3.1. | {y ∈ G : y ≺ x and {x, y} ∈ E} | < µ for all x ∈ G.
⊢Suppose x ∈ G and let α < κ be such that x ∈ Gα+1 \ Gα. Then we have
{y ∈ G : y ≺ x and {x, y} ∈ E}= ExGα ∪ {y ∈ Gα+1 \ Gα : y ≺α x and {x, y} ∈ E}.
Since | ExGα |, | {y ∈ Gα+1 \ Gα : y ≺α x} | < µ by the assumption of the lemmaand the choice of ≺α, it follows that | {y ∈ G : y ≺ x and {x, y} ∈ E} | < µ.⊣ (Claim 3.3.1)
By the deﬁnition of ≺, it is clear that ≺ is of order-type λ.(Lemma 3.3)
Lemma 3.4 (Erd˝os and Hajnal [3]). For a cardinal µ and a graph G = ⟨G, E⟩,the following are equivalent:
(a)col(G) ≤ µ;
(b)there exists a mapping f : G → [G]<µ satisfying (3.2);
(c)there is a well ordering ≺ of G of order-type | G | such that
(3.3)| {y ∈ G : y ≺ x and {x, y} ∈ E} | < µ for all x ∈ G.
Proof. “(c) ⇒ (a)” is trivial. We show “(a) ⇒ (b)” and “(b) ⇒ (c)”.“(a) ⇒ (b)”: Suppose that col(G) ≤ µ and ≺ is a well-ordering of G wit-nessing this. Let f : G → [G]<µ be deﬁned by
f(x) = {y ∈ G : y ≺ x and {x, y} ∈ E}
for x ∈ G. Then f clearly satisﬁes (3.2).
“(b) ⇒ (c)”: It is enough to prove that the following (3.4)λ holds for allcardinal λ by induction on λ:
(3.4)λ for any graph G = ⟨G, E⟩ of cardinality λ with a mapping f : G → [G]<µ
satisfying (3.2), there is a well-ordering ≺ on G of order-type | G | suchthat (3.3) holds.
For λ ≤ µ, this is trivial (for a graph G = ⟨G, E⟩ of cardinality µ, any well-ordering of G of order-type µ will do).Now, assume that λ > µ and we have proved (3.4)λ′ for all λ′ < λ. Supposethat G = ⟨G, E⟩ is a graph of cardinality λ and f : G → [G]<µ satisﬁes (3.2).Let ⟨Gα : α < κ⟩ be a ﬁltration of G for κ = cf(λ) such that each Gα,α < cf(λ) is closed with respect to f. By the induction hypothesis col(Gα) ≤ µfor all α < κ. | ExGα | < µ for all α < κ and x ∈ Gα+1 by Lemma 3.2. By Lemma3.3, it follows that there is a well-ordering ≺ of G of order-type | G | satisfying(3.3).(Lemma 3.4)
Proposition 3.5. Suppose that FRP does not hold and let
λ∗ = min{λ : λ is regular and FRP(λ) does not hold}.
Then there is a graph G = ⟨G, E⟩ of cardinality λ∗ such that col(G) > ℵ0 butall subgraphs of G of cardinality < λ∗ have countable coloring number.
Proof. By Proposition 2.6, there are a stationary E∗ ⊆ Eλ∗ωand an almostessentially disjoint ladder system g∗ on E∗. Without loss of generality, we mayassume that g∗(α) consists of successor ordinals for all α ∈ E∗ since, e.g., wemay replace g∗ by g∗∗ deﬁned by g∗∗(α) = {ξ + 1 : ξ ∈ g∗(α)} for all α ∈ E∗.Let D∗ = λ∗ \ Lim(λ∗) and let G = E∗ ∪ D∗ be the graph with its edges Edeﬁned by
(3.5)E = {{α, β} : α < β < λ∗, α ∈ D∗, β ∈ E∗ and α ∈ g∗(β)}.
We show in the following two claims that this G = ⟨G, E⟩ is as desired.
Claim 3.5.1. col(G) > ℵ0.
Proof. Otherwise there would be a mapping f : G → [G]<ℵ0 satisfying (3.2).We may assume that f(α) ⊆ g∗(α) ⊆ D∗ ∩ α for all α ∈ E∗ .Let f ∗ : E∗ → λ∗ be the regressive function deﬁned by f ∗(α) = max f(α)+1for α ∈ E∗. By Fodor’s Lemma, there are a ξ∗ < λ∗ and a stationary E′ ⊆ E∗
such that f ∗(α) = ξ∗ for all α ∈ E′. Similarly, there are a ξ∗∗ < λ and astationary E′′ ⊆ E′ such that min(g∗(α) \ ξ∗) = ξ∗∗ for all α ∈ E′′.Letα∗ ∈ E′′ \ f(ξ∗∗). Then {ξ∗∗, α∗} ∈ E but α∗ ̸∈ f(ξ∗∗) and ξ∗∗ ̸∈ f(α∗). This isa contradiction.⊣ (Claim 3.5.1)
Claim 3.5.2. col(G ↾ β) ≤ ℵ0 for every β < λ∗.⊢For β < λ∗, let f ∗ : E∗ ∩ β → λ∗ be a regressive function witnessing theessential disjointness of g∗ ↾ (E∗ ∩ β). Let f : G ∩ β → [G ∩ β]<ℵ0 be deﬁned by
f(α) =

{γ},if α ∈ D∗ ∩ β and α ∈ g∗(γ) \ f ∗(γ)for some γ ∈ E∗ ∩ β;
g∗(α) ∩ f ∗(α),if α ∈ E∗ ∩ β;
∅,otherwise
for α ∈ G ∩ β. Then f satisﬁes (3.2).⊣ (Claim 3.5.2)(Proposition 3.5)The following theorem is going to be used in the proof of Theorem 3.7.Though it can be obtained as a corollary of Shelah’s Singular CompactnessTheorem ([22]), we give below a direct proof for completeness. Note howeverthat the following Theorem 3.6 does not cover everything about coloring numberobtained from Shelah’s Singular Compactness Theorem: It also proves Theorem3.6 for arbitrary inﬁnite κ in place of ℵ0 while our proof does not cover thesecases.
Theorem 3.6 (A consequence of the main result in S. Shelah, [22]). Supposethat G = ⟨G, E⟩ is a graph of singular cardinality λ. If
(3.6)col(G′) ≤ ℵ0 for all G′ ∈ [G]<λ,
then col(G) ≤ ℵ0.
Proof. We prove the theorem in the following two cases:
Case I: cf(λ) = ω.
Let ⟨Gα,n : n < ω⟩ and ⟨fα,n : n < ω⟩ for α < ω1 be sequences constructed byinduction on α < ω1 such that
(3.7)⟨Gα,n : n < ω⟩ is a ﬁltration of G;
(3.8)fα,n : Gα,n+1 → [Gα,n+1]<ℵ0 and fα,n is a witness of col(Gα,n+1) ≤ ℵ0(which holds by the assumption on G) in the sense of Lemma 3.4, (b);
(3.9)For each n < ω, ⟨Gα,n : α < ω1⟩ is a continuously increasing sequence;
(3.10) Gα+1,n is the closure of Gα,n in Gα,n+1 with respect to fα,n.
(3.11) Gn = ∪
α<ω1 Gα,n for each n < ω.
⟨Gn : n < ω⟩ is then a ﬁltration of G. By assumption on G, we have col(Gn) ≤ℵ0 for all n < ω.
Claim 3.6.1. For any n < ω and x ∈ Gn+1 \ Gn, we have | ExGn | < ℵ0.
⊢Let α∗ < ω1 be such that x ∈ Gα∗,n+1 \ Gα∗,n.Then x ∈ Gα,n+1 forall α∗ ≤ α < ω1.Since ⟨Gα,n : α∗ ≤ α < ω1⟩ is an increasing sequence,⟨ExGα,n : α∗ ≤ α < ω1⟩ is also increasing. By (3.10) and Lemma 3.2, ExGα,n forα∗ ≤ α < ω1 are ﬁnite sets. Hence there is α∗ ≤ α∗∗ < ω1 and a ﬁnite set esuch that ExGα,n = e for all α∗∗ ≤ α < ω. By (3.11), it follows that ExGn = e.⊣ (Claim 3.6.1)
By Lemma 3.3, it follows that col(G) ≤ ℵ0.
Case II. cf(λ) > ω.
Let κ = cf(λ) and ⟨Gα : α < κ⟩ be a ﬁltration of G. By the assumption, wehave col(Gα) ≤ ℵ0 and thus there is a fα : Gα → [Gα]<ℵ0 which is a witness ofthis in the sense of Lemma 3.4 (b) for each α < κ.Let ⟨Hβ : β < κ⟩ be (possibly) another ﬁltration of G such that each Hβis closed with respect to fα for all α < κ. Note that we can construct such aﬁltration since κ < λ. By the assumption of the theorem, we have col(Hβ) ≤ ℵ0for all β < κ.
Claim 3.6.2. For all β < κ and x ∈ G \ Hβ, we have | ExHβ | < ℵ0.
⊢Suppose that x ∈ Gα∗ for some α∗ < κ. Hβ ∩Gα is closed with respect tofαfor all α < κ by the construction of ⟨Hβ : β < κ⟩. By Lemma 3.2, it followsthat ⟨ExHβ∩Gα : α∗ ≤ α < κ⟩ is an increasing sequence of ﬁnite subsets of G.Hence there is an ordinal α∗ ≤ α∗∗ < κ and an e ∈ [G]<ℵ0 such that ExHβ∩Gα = efor all α∗∗ ≤ α < κ. Since Hβ = ∪
α<κ Hβ ∩ Gα, it follows that ExHβ = e.⊣ (Claim 3.6.2)
By Lemma 3.3, it follows that col(G) ≤ ℵ0.(Theorem 3.6)
In Fleissner [4], the assertion of the following theorem was proved underAxiom R:
Theorem 3.7. (FRP) For any graph G = ⟨G, E⟩, if
(3.12) col(G ↾ I) ≤ ℵ0 holds for all I ∈ [G]≤ℵ1,
then col(G) ≤ ℵ0.
Proof. We prove by induction on λ that the following (3.13)λ holds for allcardinals λ:
(3.13)λ For any graph G = ⟨G, E⟩ of cardinality λ, if (3.12) holds, thencol(G) ≤ ℵ0.
For λ ≤ ℵ1, (3.13)λ trivially holds.Suppose that λ > ℵ1 and we have proved (3.13)λ′ for all λ′ < λ.If λ is singular, and G is as in (3.13)λ, then we can conclude col(G) ≤ ℵ0by the induction hypothesis and Theorem 3.6.Suppose now that λ is regular and assume, toward a contradiction, that thereis a graph G of cardinality λ which satisﬁes (3.12) but col(G) > ℵ0. Withoutloss of generality, we may assume that (the underlying set of) G is λ. Note thatcol(G ↾ α) ≤ ℵ0 for all α < λ by induction hypothesis. Hence
(3.14) E = {α ∈ λ : there is β ∈ λ \ α such that | Eβα | ≥ ℵ0}.
is stationary by Lemma 3.3. Let E∗ = E ∩ Eλω.
Claim 3.7.1. E∗ is stationary in λ.
⊢Suppose otherwise. Then E∩Eλ>ω must be stationary. For each α ∈ E∩Eλ>ω,let βα ∈ λ \ α be such that Eβααis inﬁnite. Let cα ∈ [Eβαα ]ℵ0 and ξα = sup(cα).Then ξα < α since cf(ξα) ≤ ω. βα and cα witness that [ξα, α) ∩ Eλω ⊆ E∗. ByFodor’s Lemma, there are ξ∗ ∈ Eλω and stationary E† ⊆ E ∩ Eλ>ω such thatξα = ξ∗ for all α ∈ E†. We have Eλω \ ξ∗ = ∪
α∈E†[ξ∗, α) ∩ Eλω ⊆ E∗. Thus E∗ isstationary. This is a contradiction to the assumption.⊣ (Claim 3.7.1)
For α ∈ E∗, let βα ∈ λ \ α be such that | Eβαα | ≥ ℵ0 and cα ∈ [Eβαα ]ℵ0. Letg : E∗ → [λ]ℵ0 be deﬁned by g(α) = cα ∪ {βα} for all α ∈ E∗.By (the original version of) FRP(λ) and Lemma 2.1, there is I ∈ [λ]ℵ1 suchthat cf(I) = ω1, I is closed with respect to g and Z as in Lemma 2.1, (c) (withE there replaced by E∗) is stationary in [I]ℵ0.
Now, we have col(G ↾ I) ≤ ℵ0 by the assumption (3.12). Hence there is awitness f : I → [I]<ℵ0 of this inequality in the sense of Lemma 3.4, (b). LetC = {x ∈ [I]ℵ0 : x is closed with respect to f}. Since C is a club in [I]ℵ0,there is an x ∈ Z ∩ C. By the deﬁnition of Z and g, | Eβx | ≥ ℵ0 for β = βsup(x).But since x is closed with respect to f, this is a contradiction to Lemma 3.2.
(Theorem 3.7)
Now we obtain Theorem 3.1 combining the results above:Proof of Theorem 3.1: “FRP(< λ) ⇒ (D)” is just the local but cumulativeversion of Theorem 3.7. “(D) ⇒ FRP(< λ)” follows from Proposition 3.5.
(Theorem 3.1)
4Collectionwise Hausdorﬀ spaces
In this section, we always assume that topological spaces satisfy T1.A topological space X is said to be collectionwise Hausdorﬀ (cwH, for short)if, for any closed and discrete D ⊆ X, there is a family U of pairwise disjointopen sets such that, for all d ∈ D, there is U ∈ U with D ∩ U = {d}. For Dand U as above, we say that D is simultaneously separated by U. We also saythat D is simultaneously separated if it is simultaneously separated by some U.X is ≤ λ-cwH if every subspaces Y of X of size ≤ λ are cwH.A topological space X has local density ≤ κ, if for every p ∈ X, there is aY ∈ [X]≤κ such that p ∈ int(Y ).The main theorem of this section is the further characterization of FRP(< λ)now in terms of collectionwise Hausdorﬀ spaces:
Theorem 4.1. For a cardinal λ ≥ ℵ2, FRP(< λ) is equivalent to the followingassertion:
(E) For every countably tight topological space X of local density ≤ ℵ1, ifX is ≤ ℵ1-cwH, then every closed discrete subsets of X of size < λ aresimultaneously separated.
The next corollary is an immediate consequence of Theorem 4.1:
Corollary 4.2. The following are equivalent:
(a)FRP;
(b)For every countably tight topological space X of local density ≤ ℵ1, if X is≤ ℵ1-cwH, then X is cwH.
Fleissner [4] proved the assertion (b) of Corollary 4.2 under Axiom R.We need the following lemma for the beginning of the induction proof of“FRP(< λ) ⇒ (E)” in Theorem 4.1.
Lemma 4.3. Suppose that X is a space having local density ≤ κ for an un-countable cardinal κ. If X is ≤ κ-cwH, then every closed discrete subsets of Xof cardinality ≤ κ are simultaneously separated.
Proof. Suppose that X and κ are as above and D ∈ [X]≤κ is closed anddiscrete. Let ⟨pβ : β < µ⟩ be an enumeration of D with µ = | D | ≤ κ and letYβ ∈ [X]≤κ, β < µ be such that pβ ∈ Uβ for Uβ = int(Yβ) for each β < µ.Let Y = D ∪ ∪
β<µ Yβ. Then | Y | ≤ κ. Since X is ≤ κ-cwH, the subspaceY of X is cwH. Since D is closed and discrete in Y , there is a sequence ⟨Oβ :β < µ⟩ of open sets in X such that pβ ∈ Oβ for all β < µ and
(4.1)Oβ ∩ Y , β < µ are pairwise disjoint.
Claim 4.3.1. Oβ ∩ Uβ, β < µ are pairwise disjoint.
⊢Suppose that (Oβ ∩ Uβ) ∩ (Oβ′ ∩ Uβ′) ̸= ∅ for some β < β′ < µ. SinceOβ ∩ Yβ is dense in Oβ ∩ Uβ, there is an x ∈ (Oβ ∩ Yβ) ∩ (Oβ′ ∩ Uβ′). But thenx ∈ (Oβ ∩ Y ) ∩ (Oβ′ ∩ Y ). This is a contradiction to (4.1).⊣ (Claim 4.3.1)
Thus ⟨Oβ ∩ Uβ : β < µ⟩ simultaneously separates D in X.(Lemma 4.3)
The following “Singular Compactness Theorem” is also used as a part ofthe proof of “FRP(λ) ⇒ (E)” of Theorem 4.1. For some other similar singularcompactness results on collectionwise Hausdorﬀ spaces, see e.g. Watson [27].
Theorem 4.4. Let λ be a singular cardinal with µ = cf(λ) < λ. Suppose thatX is a topological space of local density ≤ κ for some κ < λ. If
(4.2)every closed discrete subsets of X of cardinality < λ are simultaneouslyseparated,
then every closed discrete subsets of X of cardinality ≤ λ are simultaneouslyseparated.
Proof. Suppose that λ, µ, X and κ are as above and D ∈ [X]λ is closed anddiscrete in X. Let ⟨Di : i < µ⟩ be a ﬁltration of D. Note that, since D isclosed and discrete, each Di is closed and discrete in X.Since each Di is of cardinality λi < λ, Di is simultaneously separated by afamily Ui of pairwise disjoint open sets by (4.2).Since X is of local density ≤ κ, we may assume that
(4.3)each U ∈ Ui has a dense subset DU of size ≤ κ.
Let U = ∪
i<µ Ui and let ≈ be the transitive closure of the relation U ∼ V⇔ U ∩ V ̸= ∅.By (4.3), we have | {V ∈ U : U ∼ V } | ≤ κ+µ < λ for all U ∈ U. It followsthat | {V ∈ U : U ≈ V } | ≤ κ + µ < λ for all U ∈ U.Let E be the set of all equivalence classes of ≈. For e ∈ E, let Oe = ∪ e andD′e = D ∩ Oe. Then Oe, e ∈ E are pairwise disjoint open sets, ∪
e∈E D′e = Dand | D′e | ≤ κ + µ < λ for all e ∈ E. Since each D′e for e ∈ E is a closed discretesubset of X, D′e is simultaneously separated by some family U′e of pairwisedisjoint open sets by (4.2). We may assume that ∪ U′e ⊆ Oe for all e ∈ E. Butthen U′ = ∪
e∈E U′e simultaneously separates D.(Theorem 4.4)
Proof of Theorem 4.1: “(E) ⇒ FRP(< λ)”: By Theorem 2.7, it is enoughto show the following. Note that ADS−(µ) for a regular uncountable cardinalµ implies that µ ≥ ℵ2.
Claim 4.1.1. Suppose that ADS−(µ) holds for a regular uncountable cardinalµ < λ. Then there exists a space X of cardinality µ such that X is locallycountable, ﬁrst countable and < µ-cwH but not cwH.
⊢Let g : S → [µ]ℵ0 be an almost essentially disjoint ladder system on somestationary S ⊆ Eµω. We may assume that g(α) ∩ S = ∅ for all α ∈ S.Let T = ∪{g(α) : α ∈ S} and let X = S ∪ T be the topological space withthe topology deﬁned by:
(4.4)Every β ∈ T are isolated;
(4.5)For α ∈ S, {{α} ∪ (g(α) \ β) : β < α} is a neighborhood base of α.
Then it is easy to see that this X is as desired.⊣ (Claim 4.1.1)
“FRP(< λ) ⇒ (E)”: Suppose that X is a countably tight ≤ ℵ1-cwH spaceof local density ≤ ℵ1. By induction on κ < λ, we show that
(∗)κ if D ∈ [X]≤κ is closed and discrete then D can be simultaneously sep-arated.
For κ ≤ ℵ1, (∗)κ holds by Lemma 4.3.Assume now that ℵ1 < κ < λ and that we have shown (∗)µ for all µ < κ.
Case I. κ is regular.Suppose that D ∈ [X]κ is closed and discrete. We have to show that D issimultaneously separated.
For each p ∈ X, let Ep ∈ [X]≤ℵ1 be such that p ∈ int(Ep). Let Y ∈ [X]κ besuch that D ⊆ Y and Y is closed with respect to the mapping p �→ Ep. Thatis, Ep ⊆ Y holds for all p ∈ Y . Let ⟨Yα : α < κ⟩ be a ﬁltration of Y such that,for all α < κ,
(4.6)Yα is closed with respect to the mapping p �→ Ep.
Without loss of generality, we may assume that Y = κ.
Claim 4.1.2. C = {α < κ : Yα ∩ D = Yα ∩ D} contains a club.
⊢Suppose, toward a contradiction, that κ\C is stationary. Since κ is regular,{α < κ : Yα = α} is club. By the countable tightness of X, it follows that
S = {α ∈ Eκω : Yα = α, D ∩ Yα ̸= D ∩ Yα}
is stationary.For α ∈ S, let pα ∈ (D ∩ Yα) \ Yα. Again by the countable tightness of X,there is cα ∈ [Yα]ℵ0 such that pα ∈ cα for all α ∈ S.Now let g : S → [κ]ℵ0 be such that g(α) = cα ∪{βα} for α ∈ S where βα < κis such that pα ∈ Yβα. By FRP(κ), there is I ∈ [κ]ℵ1 such that cf(I) = ω1, I isclosed with respect to g and
(4.7)for every f : S ∩ I → κ with f(α) ∈ g(α) ∩ α for all α ∈ S ∩ I, thereexits ξ < κ such that {α ∈ S ∩ I : f(α) = ξ} is unbounded in sup(I).
Let E = {pα : α ∈ S ∩ I}. Since E ⊆ D, E is closed and discrete. Since| E | ≤ ℵ1, E is simultaneously separated by a pairwise disjoint family U ofopen sets by Lemma 4.3. For α ∈ S ∩I, let Uα ∈ U be such that pα ∈ Uα. Sincepα ∈ cα, cα ∩ Uα ̸= ∅. Let f(α) ∈ cα ∩ Uα for all α ∈ S ∩ I. By (4.7), there areα, α′ ∈ S ∩ I such that
(4.8)α < βα < α′ and
(4.9)f(α) = f(α′).
By the deﬁnition of pα’s and (4.8), we have pα ̸= pα′ and thus Uα ̸= Uα′ andUα ∩ Uα′ = ∅ by the choice of U. On the other hand, we have f(α) ∈ Uα ∩ Uα′by (4.9). This is a contradiction.⊣ (Claim 4.1.2)
By moving to a subsequence of ⟨Yα : α < κ⟩ with the club subset of κ inClaim 4.1.2 as the new index set, we may assume that
(4.10) D ∩ Yα = D ∩ Yα for all α < κ.
Claim 4.1.3. D ∩ Yα = D ∩ int(Yα) for all α < κ
⊢“⊇” follows from (4.10). For “⊆”, if p ∈ D ∩ Yα, then Ep ⊆ Yα by (4.6),and p ∈ int(Ep) ⊆ int(Yα). Thus p ∈ D ∩ int(Yα).⊣ (Claim 4.1.3)
For α < κ, let Dα = (D ∩ Yα+1) \ Yα. Since | Dα | < κ, Dα is simultaneouslyseparated by a pairwise disjoint family Uα of open sets. By (4.10) and Claim4.1.3, we may assume that U ⊆ int(Yα+1)\Yα for all U ∈ Uα. Then U = ∪
α<κ Uαseparates D simultaneously.
Case II. κ is singular.In this case, if D ∈ [X]κ is closed and discrete, then it is simultaneously sepa-rated by the induction hypothesis and Theorem 4.4.(Theorem 4.1)
By Theorem 4.1 and Claim 4.1.1, the following assertion is also equivalentto FRP(< λ):
(E’) For every locally countable, ﬁrst countable topological space X, if Xis ≤ ℵ1-cwH, then every closed discrete subsets of X of size < λ aresimultaneously separated.
5Separation of FRP from RP
As it was already mentioned in the introduction, it is fairly easy to separateFRP from RP: Any model of FRP + 2ℵ0 > ℵ2 does not satisfy WRP sinceWRP([ω2]ℵ0) implies 2ℵ0 ≤ ℵ2.It is also relatively easy to obtain models of FRP + ¬WRP([ω2]ℵ0) togetherwith 2ℵ0 ≤ ℵ2 : Suppose that κ is a strongly compact cardinal.Then thecollapsing Col(ω1, < κ) of κ to ω2 by countable conditions forces FRP (seeProposition 5.1 below).Forcing further by the partial ordering C adding aCohen real over this model still preserve FRP by Theorem 3.4 in [9]. On theother hand {x ∈ [ω2]ℵ0 : x ̸∈ V Col(ω1,<κ)} is a non-reﬂecting stationary set inthe generic extension by Col(ω1, < κ) ∗ C (see [12]). This construction givesthus a model of FRP + ¬WRP([ω2]ℵ0) + CH.Likewise, the forcing by Col(ω1, < κ)∗Cℵ2 for a strongly compact κ producesa model of FRP + ¬WRP([ω2]ℵ0) + 2ℵ0 = ℵ2.In the rest of this section, we give two constructions of models of FRP+ ¬WRP([ω2]ℵ0) + 2ℵ0 ≤ ℵ2 without adding reals.The ﬁrst construction(Theorem 5.2) relies on the model of set theory given in Section 5 of Sakai [19].First let us check the following:
Proposition 5.1. Suppose that κ ≤ λ, κ is λ-strongly compact and λ is regular.Then ∥– P “ FRP(λ) ” holds for P = Col(ℵ1, < κ).
Proof. Let j : V ≼ M be such that
(5.1)crit(j) = κ;
(5.2)M κ ⊆ M and
(5.3)∀X ∈ [M]≤λ ∃Y ∈ ([M]<j(κ))M (X ⊆ Y )
(see e.g. Theorem 22.17 in [13]). Suppose that ˙S is a P-name of a stationarysubset of Eλω and ˙g a P-name such that
(5.4)∥– P “ ˙g : S → [λ]ℵ0 and ˙g is a ladder system on ˙S ”.
Let P′ = Col(ℵ1, < j(κ)). Thus we have P′ = j(P) by (5.2). Let G′ be a(V, P′)-generic set such that G ⊆ G′. j can be then extended to the elementaryembedding j′ : V [G] ≼ M[G′] deﬁned by j′(˙aG) = j(˙a)G′ for each P-name ˙a.In V [G], let
˜S = {x ∈ [λ]ℵ0 : sup(x) ∈ ˙SG and ˙gG(sup(x)) ⊆ x}.
Then ˜S is a stationary subset of [λ]ℵ0. Note that here we need the regularity ofλ. Since P′ is σ-closed, ˜S remains stationary in V [G′]. It follows that
(5.5)j′ ′′ ˜S is a stationary subset of [j ′′λ]ℵ0 in V [G′].
Let α∗ = sup(j ′′λ). Note that V [G′] |= cf(α∗) = ω1. Hence M[G′] |= cf(α∗) >ω.By M[G′] |= α∗ ≤ j(λ) and (5.3), we have M[G′] |= cf(α∗ < j(κ) =ω2). Hence M[G′] |= cf(α∗) = ω1. By (5.3), there is Y ∈ M such that Y ∈([α∗]<j(κ))M and j ′′λ ⊆ Y . In M[G′], let ˜Y be the closure of Y with respect toj′(˙gG). Since M[G′] |= j(κ) = ω2, we have M[G′] |= ˜Y ∈ [α∗]ℵ1.In M[G′], let ⟨Yα : α < ω1⟩ be a ﬁltration of ˜Y . Then ⟨Y ′α : α < ω1⟩ forY ′α = Yα ∩ j ′′λ is a ﬁltration of j ′′λ (in V [G′]). By (5.5), the set
{α < ω1 : sup(Yα) = sup(Y ′α) and j′(˙gG)(sup(Yα)) ⊆ Y ′α}
is stationary in V [G′]. It follows that {α < ω1 : j′(˙gG)(sup(Yα)) ⊆ Yα} isstationary in V [G′] and hence also in M[G′].Thus, in M[G′], we have
M[G′] |= ∃I ∈ [j(λ)]ℵ1 ( cf(sup(I)) = ω1, I is closed with respect to
j′(˙gG) and I has a ﬁltration ⟨Iα : α < ω1⟩
such that {α < ω1 : j′(˙gG)(sup(Iα)) ⊆ Iα}
is stationary ).
By the elementarity of j′, it follows that
V [G] |= ∃I ∈ [λ]ℵ1 ( cf(sup(I)) = ω1, I is closed with respect to ˙gG
and I has a ﬁltration ⟨Iα : α < ω1⟩ such that
{α < ω1 : ˙gG(sup(Iα)) ⊆ Iα} is stationary ).
Thus, by Theorem 2.2 and Lemma 2.1, we have V [G] |= FRP(λ).
(Proposition 5.1)
Theorem 5.2. Suppose that there is a supercompact cardinal κ. Then there isa partial ordering P collapsing κ to ω2 without adding any new reals such that
(5.6)∥– P “ FRP + ¬WRP([ω3]ℵ0) + 2ℵ0 = ℵ1 ”.
Note that, since WRP enjoys downward transfer property, (5.6) implies
(5.7)∥– P “ ¬WRP([λ]ℵ0) for all λ ≥ ω3 ”.
Proof of Theorem 5.2: By Theorem 5.3 in [19], there is a partial orderingP0 such that P0 adds no new reals and
∥– P0 “ κ is strongly compact ” and∥– P0∗Col(ℵ1,< κ)P0 “ ¬WRP([ω3]ℵ0) ”.
On the other hand, we have ∥– P0∗Col(ℵ1,< κ)P0 “ FRP ” by Proposition 5.1.
(Theorem 5.2)
Note that the model above also satisﬁes RC (the proof is similar to andmuch simpler than that of Proposition 5.1). Thus we can conclude that RCdoes not imply WRP. This solves the Question 7.9 in [25] negatively.In Theorem 5.2, ℵ3 is the ﬁrst cardinal κ where the consistence of FRPtogether with ¬WRP([κ]ℵ0) is shown. T. Miyamoto [18] proved that the consis-tency strength of FRP(ℵ2) is exactly that of the existence of a Mahlo cardinal.His construction also gives a model of FRP(ω2) + ¬WRP([ω2]ℵ0) + CH. In thefollowing, we construct of a model of FRP and ¬WRP([ℵ2]ℵ0) by forcing witha σ-Baire partial ordering over a model with a strongly compact cardinal.Let us begin with a well-known fact (Fact 5.3) which plays the central rolein the following Theorem 5.4 as well as in Theorem 6.1 in the next section. Forcompleteness, we shall give a proof of it. The following proof is a variant of theargument with a game considered in Veliˇckovi´c [26].
Fact 5.3. Let λ be an ordinal with cf(λ) > ω1. For any stationary E ⊆ Eλωand any mapping f : E → [λ]ℵ0 such that f(α) is a coﬁnal subset of α for allα ∈ E, the set
Tf = {x ∈ [λ]ℵ0 : f(sup(x)) ̸⊆ x}
is stationary in [λ]ℵ0.
Proof. Suppose that C ⊆ [λ]ℵ0 is a club. We want to show that Tf ∩ C ̸= ∅.Let g : λ<ω → λ be a mapping such that Cg ⊆ C where Cg = {x ∈ [λ]ℵ0 :x is closed with respect to g}. It is enough to show that Tf ∩ Cg ̸= ∅.Let ⅁(g) be the following game of length ω for the players I and II: A matchM in ⅁(g) looks like this:
M:Player Iα0α1· · ·αn· · ·
Player III0, ξ0I1, ξ1· · ·In, ξn· · ·(n < ω)
where each In is a closed interval [βn, γn] and
(5.8)αn < βn ≤ ξn < γn < αn+1
for all n ∈ ω.Player II wins the match if cl g({ξn : n ∈ ω}) ⊆ γ0 ∪ ∪
n∈ω\1 In.Herecl g(s) for s ⊆ λ denotes the closure of the set s with respect to g. That is, the⊆-minimal set s′ ⊆ λ such that s ⊆ s′ and g(u) ∈ s′ for all u ∈ s′<ω.The proof of the following claim is an adaptation of the corresponding proofin [26]:
Claim 5.3.1 (Veliˇckovi´c). II has a winning strategy in ⅁(g).
We ﬁrst show that the claim above implies the theorem. Without loss ofgenerality, we may assume that f is a ladder system on E. Let σ be a winningstrategy of II in ⅁(g). Let θ be a suﬃciently large regular cardinal and letM ≺ ⟨H(θ), ∈⟩ be such that
(5.9)M is countable;
(5.10) g, σ ∈ M and
(5.11) δ = sup λ ∩ M ∈ E.
M∗:Player Iα0α1· · ·αn· · ·
Player III0, ξ0I1, ξ1· · ·In, ξn· · ·
be a match in ⅁(g) such that
(5.12) II played according to σ in M∗;
(5.13) each initial segment ⟨α0, β0, ξ0, γ0, ..., αn, βn, ξn, γn⟩ of M∗ is in M and
(5.14) the interval (γn, αn+1), contains an element δn of f(δ) for n ∈ ω.
Let x = cl g({ξn : n ∈ ω}). Then x ∈ Cg. sup(x) = δ by (5.13) and (5.14). By(5.12) and (5.14) we have x ∩ {δn : n ∈ ω} = ∅. Since {δn : n ∈ ω} ⊆ f(δ)by (5.14), it follows that f(sup(x)) ̸⊆ x and hence x ∈ Tf. This shows thatTf ∩ Cg ̸= ∅ as desired.
⊢(Claim 5.3.1) Suppose otherwise. Since ⅁(g) is an open game for the player I,the player I then has a winning strategy σ by Gale-Stewart Theorem.Let⟨δn : n ∈ ω⟩ be a strictly increasing sequence of ordinals below λ such that
(5.15) cf(δn) = ω1 and
(5.16) δn is closed with respect to g and σ for all n ∈ ω.
Note that we can take such sequence by cf(λ) > ω1.Let δ = supn∈ω δn. Let x ∈ [δ]ℵ0 be such that
(5.17) sup(x) = δ;
(5.18) [δn, δn + ω] ⊆ x for all n ∈ ω and
(5.19) x is closed with respect to σ and g.
Let αn, βn, ξnγn ∈ δ for n ∈ ω be such that
(5.20) αn < βn = ξn < γn < αn+1;
(5.21) αn = σ(⟨αk, βk, ξk, γk : k < n⟩);
(5.22) βn = δn, ξn = δn and
(5.23) γn = sup(δn+1 ∩ x)
for all n ∈ ω.By (5.15) and (5.23) we have γn < δn+1. If ⟨αk, βk, ξk, γk : k < n⟩ is asequence below δn satisfying (5.20) then, since δn is closed with respect to σ,we have
(5.24) αn < δn.
Thus the construction according to (5.21), (5.22), (5.23) provides a sequencesatisfying (5.20).Letting In = [βn, γn] for n ∈ ω, the sequence
α0, I0, ξ0, α1, I1, ξ1,..., αn, In, ξn,...
is a match in ⅁(g) in which the player I played according to the winning strategyσ. However, by (5.19) and (5.23), we have
cl g({ξn : n ∈ ω}) ⊆ x ⊆ γ0 ∪ ∪
n∈ω\1 In.
Thus Player II wins this match. This is a contradiction.⊣ (Claim 5.3.1)
(Fact 5.3)
Theorem 5.4. Suppose that κ is a strongly compact cardinal. Then there existsa κ-c.c. and σ-Baire forcing notion P which forces
(5.25) κ = ω2, CH and FRP but ¬WRP([ω2]ℵ0).
Proof. The idea of the proof is that we introduce a partial ordering P suchthat a (V, P)-generic set G collapses κ to ω2 without adding new countable setand it introduces a mapping fG and gG such that fG maps each α ∈ Eκω to acoﬁnal countable subset of α while gG prevents TfG deﬁned as in Fact 5.3 frombecoming a reﬂecting stationary set. This destroys WRP([ω2]ℵ0) in the genericextension. At the same time, we design P nice enough to preserve stationarityof certain type of sets so that a proof similar to but slightly more complicatedthan that of Theorem 5.1 works for this P to show that P forces FRP.Let P be the set of all pairs ⟨f, g⟩ of functions such that
(5.26) dom(f) ∈ [Eκω]≤ℵ0,
(5.27) f(α) ∈ [α]ℵ0 and f(α) is coﬁnal in α for all α ∈ dom(f),
(5.28) dom(g) ∈ [Eκ>ω]≤ℵ0,
(5.29) g(β) ∈ [[β]ℵ0]≤ℵ0 and, with respect to ⊆, g(β) is well-ordered, closed(that is, the corresponding increasing sequence of countable sets is con-tinuous) and with the maximal element,
(5.30) sup(x) ∈ dom(f) for every x ∈ ∪{g(β) : β ∈ dom(g)}, and
(5.31) f(sup(x)) ⊆ x for every x ∈ ∪{g(β) : β ∈ dom(g)}.
For ⟨f, g⟩, ⟨f ′, g′⟩ ∈ P, we deﬁne
(5.32) ⟨f, g⟩ ≤P ⟨f ′, g′⟩ ⇔ f ′ ⊆ f, dom(g′) ⊆ dom(g) and g(β) isan end extension of g′(β) for all β ∈ dom(g′).
In the following, we show that this ⟨P, ≤P⟩ is as desired.
Claim 5.4.1. For any x∗ ∈ [κ]ℵ0,
(5.33) Dx∗ = {⟨f, g⟩ ∈ P : (a) x∗ ∩ Eκω ⊆ dom(f),
(b) x∗ ∩ Eκ>ω ⊆ dom(g),
(c) ∀β ∈ dom(g) ∃y ∈ g(β) (x∗ ∩ β ⊆ y) }
is dense in P.
⊢Let θ be a suﬃciently large regular cardinal. For an arbitrary ⟨f ′, g′⟩ ∈ P,let N ≺ H(θ) be countable such that x∗, ⟨f ′, g′⟩ ∈ N.Let ⟨f, g⟩ be the pair of the functions deﬁned by
(5.34) dom(f) = (Eκω ∩ N) ∪ Lim(κ ∩ N) and
f(α) =
{f ′(α),if α ∈ dom(f ′),
α ∩ N,if α ∈ dom(f) \ dom(f ′)
for α ∈ dom(f);
(5.35) dom(g) = Eκ>ω ∩ N and
g(β) =
{g′(β) ∪ {β ∩ N},if β ∈ dom(g′),
{β ∩ N},if β ∈ dom(g) \ dom(g′)
for β ∈ dom(g).
⟨f, g⟩ satisﬁes (a), (b), (c) of (5.33). Since the pair ⟨f, g⟩ and ⟨f ′, g′⟩ alsosatisﬁes the conditions in the deﬁnition (5.32) of ≤P, it is enough to show that⟨f, g⟩ ∈ P. That ⟨f, g⟩ satisﬁes (5.26) ∼ (5.29) is clear.For (5.30) and (5.31), suppose that x ∈ g(β) for some β ∈ dom(g) ⊆ Eκ>ω. Ifβ ∈ dom(g′) and x ∈ g′(β) then sup(x) ∈ dom(f ′) ⊆ dom(f) and f(sup(x)) =f ′(sup(x)) ⊆ x. Otherwise x = β ∩ N by (5.35). Then sup(x) ∈ Lim(β ∩ N) ⊆dom(f) and, by (5.34), f(sup(x)) = sup(x) ∩ N = x.⊣ (Claim 5.4.1)
Claim 5.4.2. P is σ-Baire.
⊢Suppose that Di, i ∈ ω are dense open sets in P and ⟨f ′, g′⟩ ∈ P. We showthat there is ⟨f, g⟩ ∈ P such that ⟨f, g⟩ ≤P ⟨f ′, g′⟩ and ⟨f, g⟩ ∈ Di for all i ∈ ω.Let θ be a suﬃciently large regular cardinal and N ≺ ⟨H(θ), ∈⟩ be countablesuch that P, ⟨Di : i ∈ ω⟩, ⟨f ′, g′⟩ ∈ N.Let ⟨xi : i ∈ ω⟩ be an increasing sequence in [κ]ℵ0 ∩ N such that
(5.36) ∪
i<ω xi = κ ∩ N.
By Claim 5.4.1, by closedness of Di’s and by the elementarity of N, we can ﬁnda decreasing sequence ⟨fi, gi⟩, i ∈ ω in P such that
(5.37) ⟨f0, g0⟩ ≤P ⟨f ′, g′⟩;
for all i ∈ ω,
(5.38) ⟨fi, gi⟩ ∈ Di ∩ N, and
(5.39) Eκω ∩ xi ⊆ dom(fi), Eκ>ω ∩ xi ⊆ dom(gi) and xi ∩ β ⊆ ∪ gi(β) for allβ ∈ dom(gi).
By (5.36) and (5.39), we have
(5.40) ∪
i∈ω dom(fi) = Eκω ∩ N,∪
i∈ω dom(gi) = Eκ>ω ∩ Nand
(5.41) ∪{∪ gi(β) : i < ω, β ∈ dom(gi)} = β ∩ N
for all β ∈ Eκ>ω ∩ N.Let ⟨f, g⟩ be the pair of functions f, g such that
(5.42) dom(f) = (Eκω ∩ N) ∪ Lim(κ ∩ N), and
f(α) =
{fi(α),if α ∈ dom(fi) for some i ∈ ω,
N ∩ α,otherwise,
for α ∈ dom(f),
(5.43) dom(g) = Eκ>ω ∩ N, and
g(β) = ∪{gi(β) : i < ω, β ∈ dom(gi)} ∪ {β ∩ N}for all β ∈ dom(g).
Then we have ⟨f, g⟩ ∈ P: It is clear that ⟨f, g⟩ satisﬁes (5.26) ∼ (5.29). For(5.30) and (5.31), suppose that x ∈ ∪{g(β) : β ∈ dom(g)}. If x ∈ gi(β) forsome i ∈ ω and β ∈ dom(gi), then sup(x) ∈ dom(fi) and fi(sup(x)) ⊆ x, since⟨fi, gi⟩ ∈ P. It follows that sup(x) ∈ dom(f) and f(sup(x)) ⊆ x by (5.42).Otherwise, x = β ∩ N for some β ∈ dom(g) and sup(x) ∈ Lim(κ ∩ N). Thenf(sup(x)) = sup(x) ∩ N by (5.42) and we again have f(sup(x)) ⊆ x.⟨f, g⟩ ≤P ⟨f ′, g′⟩ by (5.37) and ⟨f, g⟩ ∈ Dn for all n ∈ ω by (5.38).⊣ (Claim 5.4.2)
Claim 5.4.3. Suppose ⟨f, g⟩, ⟨f ′, g′⟩ ∈ P. Then
⟨f, g⟩ and ⟨f ′, g′⟩ are compatible ⇔
(5.44) f ↾(dom(f) ∩ dom(f ′))= f ′ ↾(dom(f) ∩ dom(f ′)), and
(5.45) for every β ∈ dom(g) ∩ dom(g′), one of g(β) and g′(β) is an initialsegment of the other.
⊢“⇒ is clear by the deﬁnition of ≤P.For “⇐”, suppose that ⟨f, g⟩, ⟨f ′, g′⟩ ∈ P satisfy (5.44) and (5.45).Let⟨f ∗, g∗⟩ be deﬁned by
(5.46) f ∗ = f ∪ f ′,
(5.47) dom(g∗) = dom(g) ∪ dom(g′), and
g∗(β) =
g(β),if β ∈ dom(g) \ dom(g′),
g′(β),if β ∈ dom(g′) \ dom(g),
g(β) ∪ g′(β),if β ∈ dom(g) ∩ dom(g′)
for β ∈ dom(g∗).
Then ⟨f ∗, g∗⟩ ∈ P and ⟨f ∗, g∗⟩ ≤P ⟨f, g⟩, ⟨f ′, g′⟩.⊣ (Claim 5.4.3)
Claim 5.4.4. P satisﬁes the κ-Knaster property. In particular, P satisﬁes theκ-c.c.
⊢Suppose A ∈ [P]κ. Then by ∆-System Lemma and Pigeon Hole Principle,we can ﬁnd r0, r1 ∈ [κ]ℵ0 and B ∈ [A]κ such that
(5.48) for any distinct ⟨f, g⟩ and ⟨f ′, g′⟩ ∈ B, we have dom(f) ∩ dom(f ′) = r0and f ↾ r0 = f ′ ↾ r0,
(5.49) for any distinct ⟨f, g⟩ and ⟨f ′, g′⟩ ∈ B, we have dom(g) ∩ dom(g′) = r1and g ↾ r1 = g′ ↾ r1 .
By Claim 5.4.3, the elements of B are pairwise compatible. ⊣ (Claim 5.4.4)
For (V, P)-generic G, let
(5.50) fG = ∪{f : ⟨f, g⟩ ∈ G for some g} and
(5.51) gG = ∪{g : ⟨f, g⟩ ∈ G for some f}.
Claim 5.4.5. In V [G], we have the following:
(1) ωV [G]1= ωV1 .
(2) fG : Eκω → [κ]ℵ0, fG(α) ∈ [α]ℵ0 and fG(α) is coﬁnal in α for all α ∈ Eκω.
(3) dom(gG) = Eκ>ω, gG(β) is a continuously increasing coﬁnal sequence in[β]ℵ0 of length ω1 with ∪ g(β) = β for all β ∈ Eκω and
(5.52) fG(sup(x)) ⊆ x for x ∈ gG(β).
(4) κ = ω2 and 2ℵ0 = ℵ1.(5) S = {x ∈ [κ]ℵ0 : fG(sup(x)) ̸⊆ x} (which is a stationary set in [κ]ℵ0 byFact 5.3) is non-reﬂecting stationary. That is, for every limit ordinal α < κ, S∩[α]ℵ0 is non-stationary in [α]ℵ0. In particular, V [G] is a model of ¬WRP([ω2]ℵ0).
⊢(1): By Claim 5.4.2.
(2) and (3) follow from the deﬁnition of ⟨P, ≤P⟩ and Claim 5.4.1. In partic-ular, (5.52) follows from (5.31).
(4) follows from (3) and (1).
(5): For α ∈ Eκω, it is clear that S ∩ [α]ℵ0 is non-stationary. For α ∈ Eκ>ω,gG(α) is a closed unbounded subset of [α]ℵ0 by (3) and gG(α) is disjoint fromS ∩ [α]ℵ0 by (5.52).⊣ (Claim 5.4.5)
In the rest of the proof, we show that FRP holds in V [G].Let λ ≥ κ be a regular cardinal and j : V → M be a λ-compact elementaryembedding satisfying (5.1), (5.2) and (5.3). By (5.2), we have P ∈ M and, by(5.1), j(p) = p for all p ∈ P. By elementarity, we have
(5.53) M |= j(P) = {⟨f, g⟩ : ⟨f, g⟩ |= (5.26)’ ∼ (5.31)’} and
(5.54) M |= j(≤P)
= {⟨⟨f, g⟩, ⟨f ′, g′⟩⟩ ∈ (j(P))2 : ⟨⟨f, g⟩, ⟨f ′, g′⟩⟩ |= (5.32)’}
where (5.26)’ ∼ (5.31)’, (5.32)’ are conditions obtained from (5.26) ∼ (5.31),(5.32) by replacing κ appearing there by j(κ). Hence by (5.2), it follows that
(5.55) j(P) = {⟨f, g⟩ : ⟨f, g⟩ |= (5.26)’ ∼ (5.31)’} and
j(≤P) = {⟨⟨f, g⟩, ⟨f ′, g′⟩⟩ ∈ (j(P))2 : ⟨⟨f, g⟩, ⟨f ′, g′⟩⟩ |= (5.32)’}.
In particular, the proof of Claim 5.4.2 also applies to ⟨j(P), j(≤P)⟩ and we canconclude that j(P) = ⟨j(P), j(≤P)⟩ is also σ-Baire. In the following we shalldenote j(≤P) also by ≤j(P).
Claim 5.4.6. j ↾ P : P → j(P); p �→ p is a complete embedding.
⊢Suppose that A ⊆ P is maximal pairwise incompatible. By Claim 5.4.4, wehave | A | < κ. Hence
(5.56) j(A) = j ′′A = A.
By the elementarity, we have
(5.57) M |= “j(A) is maximal pairwise incompatible in j(P)”.
So, by (5.56), it follows that j ′′A = A is maximal pairwise incompatible in j(P)(in V ).⊣ (Claim 5.4.6)
For ⟨f, g⟩ ∈ j(P), let
(5.58) ⟨f, g⟩ ↾ P = ⟨f ↾ κ, g ↾ κ⟩.
Note that ⟨f, g⟩ ↾ P ∈ V and hence ⟨f, g⟩ ↾ P ∈ P and ⟨f, g⟩ ↾ P ≤j(P) ⟨f, g⟩.Let Q = j(P)/G. By Claim 5.4.3, we have
(5.59) Q = {p ∈ j(P) : p is compatible with all j(r) = r ∈ G}
= {⟨f, g⟩ ∈ j(P) : ⟨f, g⟩ ↾ P ∈ G}.
Working in V [G], let E ⊆ Eλω be stationary and g : E → [λ]ℵ0 be a laddersystem. We are going to show that there is α∗ ∈ Eλω1 witnessing (2.17)λ forthese E and g.Let
(5.60) S = {x ∈ [λ]ℵ0 : sup(x) ∈ E, fG(sup(x ∩ κ)) ∪ g(sup(x)) ⊆ x}.
Then S is a stationary subset of [λ]ℵ0.
Claim 5.4.7. Q preserves the stationarity of S.
⊢Suppose that ⟨f †, g†⟩ ∈ Q and ˙h is a Q-name such that ∥– Q “ ˙h : λ<ω → λ ”.It is enough to show that there is ⟨f ∗, g∗⟩ ≤Q ⟨f †, g†⟩ and x∗ ∈ S such that⟨f ∗, g∗⟩ ∥– Q “ x∗ is closed with respect to ˙h ”.Let θ be a suﬃciently large regular cardinal and let N ≺ H(θ) be countablesuch that κ, j(κ), Q, ⟨f †, g†⟩, λ, g, ˙h ∈ N and
(5.61) λ ∩ N ∈ S.
For ⃗ξ ∈ (λ ∩ N)<ω, let
(5.62)¯D⃗ξ = {q ∈ Q : q decides ˙h(⃗ξ)}.
Note that ¯D⃗ξ is a dense open subset of Q.Let Dn, n ∈ ω be an enumeration of { ¯D⃗ξ : ⃗ξ ∈ (λ ∩ N)<ω} and let xi ∈[j(κ)]ℵ0 ∩ N, i ∈ ω be such that ∪
i∈ω xi = j(κ) ∩ N.Let ⟨⟨fi, gi⟩ : i ∈ ω⟩ be a descending sequence in Q ∩ N such that
(5.63) ⟨f0, g0⟩ ≤Q ⟨f †, g†⟩,
and, for all i ∈ ω,
(5.64) ⟨fi, gi⟩ ∈ Di,
(5.65) Ej(κ)ω∩ xi ⊆ dom(fi),
(5.66) Ej(κ)>ω ∩ xi ⊆ dom(gi),
(5.67) xi ∩ β ⊆ ∪ gi(β) for all β ∈ dom(gi).
Subclaim 5.4.7.1. ⟨⟨fi, gi⟩ : i ∈ ω⟩ has a lower bound ⟨f ∗, g∗⟩ in Q.Inparticular, ⟨f ∗, g∗⟩ ≤Q ⟨f †, g†⟩ by (5.63).
⊢Let ⟨f ∗, g∗⟩ be deﬁned by the following:
(5.68) dom(f ∗) = (Ej(κ)ω∩ N) ∪ Lim(j(κ) ∩ N) and
f ∗(α) =
fi(α),if α ∈ dom(fi) for some i ∈ ω(a)
α ∩ N,if α ̸∈ ∪
i∈ω dom(fi) = Ej(κ)ω∩ N and α > κ(b)
fG(α),otherwise(c)
for α ∈ dom(f ∗);
(5.69) dom(g∗) = Ej(κ)>ω ∩ N and
g∗(β) = ∪{gi(β) : β ∈ dom(g∗), i ∈ ω, β ∈ dom(gi)} ∪ {β ∩ N}
for β ∈ dom(g∗).
We check ﬁrst that ⟨f ∗, g∗⟩ ∈ j(P). It is clear that ⟨f ∗, g∗⟩ satisﬁes (5.26) ∼(5.29) (with κ there replaced by j(κ)): For (5.27), note that ∪
i∈ω dom(fi) =Ej(κ)ω∩N by fi ∈ N for i ∈ ω and (5.65), and fG(α) is coﬁnal in α for all α ∈ Eκω.For (5.29), note that ∪ ∪{gi(β) : β ∈ dom(g∗), i ∈ ω, β ∈ dom(gi)} = β ∩ Nby gi ∈ N for i ∈ ω, (5.66) and (5.67).To see that ⟨f ∗, g∗⟩ satisﬁes (5.30) and (5.31) (also with κ replaced by j(κ)),let x ∈ g∗(β) for some β ∈ dom(g∗).If x ∈ gi(β) for some i ∈ ω, then it is clear that this x satisﬁes the conditionsin (5.30) and (5.31) for ⟨f ∗, g∗⟩ since ⟨fi, gi⟩ ∈ Q.So assume that x ̸∈ ∪
i∈ω gi(β). Then, by (5.69), x = β ∩ N and sup(x) ∈Lim(j(κ) ∩ N) \ (Ej(κ)ω) ∩ N).If β > κ, then sup(x) > κ and f ∗(sup(x)) = β ∩ N = x, by (5.68), (b).If β = κ, then x = κ ∩ N and since λ ∩ N ∈ S, f ∗(sup(x)) = fG(sup(x)) =fG(sup(κ∩N)) ⊆ x by (5.68), (c) and (5.60) (note that x = κ∩N = (λ∩N)∩κ).If β < κ, there is a ⟨f ′, g′⟩ ∈ G such that
(5.70) sup(x) ∈ dom(f ′), β ∈ dom(g′) and
(5.71) x ⊆ ∪ g′(β)
by Claim 5.4.1. For each i ∈ ω such that β ∈ dom(gi), we have ∪ gi(β) ∈ g′(β)since ⟨fi, gi⟩ ↾ P ∈ G and ⟨fi, gi⟩ ∈ N, and by (5.71).Thus x = β ∩ N = ∪{∪ gi(β) : i ∈ ω, β ∈ dom(gi)} ∈ g′(β) by theclosedness of g′(β) (see (5.29)).It follows that f ∗(sup(x)) = fG(sup(x)) =f ′(sup(x)) ⊆ x.
By the argument above, it also follows that ⟨f ∗, g∗⟩ ↾ P ∈ G and hence⟨f ∗, g∗⟩ ∈ Q.By the deﬁnition of ⟨f ∗, g∗⟩ it is clear that ⟨f ∗, g∗⟩ ≤Q ⟨fi, gi⟩ for i ∈ ω.⊣ (Subclaim 5.4.7.1)
By (5.64), ⟨f ∗, g∗⟩ decides ˙h ↾ λ<ω ∩ N to be a mapping from λ<ω ∩ N toλ∩N. Thus, letting x∗ = λ∩N, we have x∗ ∈ S by the choice of N (see (5.61))and ⟨f ∗, g∗⟩ ∥– Q “ x∗ is closed with respect to ˙h ”.⊣ (Claim 5.4.7)
Now, let H be a (V [G], Q)-generic ﬁlter. Then V [G][H] = V [H] and the λ-compact elementary embedding j : V → H can be extended to the elementaryembedding ˜j : V [G] → M[H] by ˜j(˙aG) = (j(˙a))H for all P-name ˙a.Let α∗ = sup(j ′′λ). Note that α∗ < j(λ) since M |= cf(α∗) < j(κ) < j(λ)by (5.3) while M |= j(λ) is a regular cardinal. Since M |= cf(α∗) ≥ ω1, we haveM[H] |= cf(α∗) = ω1.By Claim 5.4.7, S is stationary in [λ]ℵ0. Since j ↾ λ : λ → α∗ sends λ coﬁnalin α and preserves limits of increasing sequences of length ω,
S∗ = {x ∈ [α∗]ℵ0 : ˜j(g)(sup(x)) ⊆ x}
⊇ {x ∈ [α∗]ℵ0 : x ⊇ z and sup(x) = sup(z) for some z ∈ ˜j ′′S}
is stationary in V [H]. Note that S∗ ∈ M[H]. Hence S∗ is also stationary inM[H]. Thus we have
M[H] |= ˜j(g) satisﬁes (2.17)j(λ).
By elementarity, it follows that
V [G] |= g satisﬁes (2.17)λ.
By Proposition 2.4, this shows that V [G] |= FRP.(Theorem 5.4)
6Separation of ORP from FRP
In this section, we prove the following theorem:
Theorem 6.1. Suppose that MA+(σ-closed) holds.Then, for every regularκ ≥ ω2, there is a κ-strategically closed partial ordering P∗ such that
(1) ∥– P∗ “ ORP(κ) + ¬FRP(κ) ”.
Furthermore, we have
(2) for all regular cardinals λ > 2<κ as well as for all regular cardinalsω1 < λ < κ, we have ∥– P∗ “ FRP(λ) ”.
By κ-strategically closedness and since P∗ can be chosen such that it collapsesall cardinals in the interval (κ, 2<κ], it follows that
(3) P∗ forces that κ is the unique regular cardinal ≥ ℵ2 where Fodor-typeReﬂection Principle does not hold. In particular, we have ∥– P∗ “ ORP ”.
Here, a partial ordering P is said to be α-strategically closed for an ordinalα if the Player II has a winning strategy in the following inﬁnite game ⅁α(P):In a match of ⅁α(P), Player I and II play elements of P forming a decreasingsequence in P:
Player Ip0p1· · ·pξ· · ·
Player IIq0q1· · ·qξ· · ·(ξ < α)
p0 ≥P q0 ≥P p1 ≥P q1 ≥P · · · ≥P pξ ≥P qξ ≥P pξ+1 ≥P · · ·
and such that, in each of the η’th innings for limit η < α, only Player II mayplay (to simplify the notation we assume pξ = qξ but this move is chosen byPlayer II for all limit ordinal ξ < α).Player II wins the match in ⅁α(P) if the game can be played in all of theξ’th innings for ξ < α.If P is α-strategically closed and λ < α is a cardinal, it is easy to see that Pdoes not add any new set of ordinals of size ≤ λ.A partial ordering P is said to be strongly α-strategically closed if PlayerII has a winning strategy in ⅁+α(P) which is deﬁned just as ⅁α(P) except thatPlayer I may begin each of the η’th innings for limit η < α.A typical example of strongly κ-strategically closed partial orderings for acardinal κ is a partial ordering P with a κ-closed dense subset D. Such a partialordering P is indeed strongly κ-strategically closed since taking moves alwaysfrom D is a winning strategy for Player II in ⅁+κ (P).In [16], α-strategical closedness and strong α-strategical closedness are calledα-game closedness and strong α-game closedness respectively.Let κ be a regular cardinal ≥ ω2.In the proof of Theorem 6.1 below,we show that the following partial ordering P∗ has the desired property underMA+(σ-closed).Let P∗ consist of all functions p such that
(6.1)dom(p) ⊆ Eκω and | p | < κ;
(6.2)p is a ladder system on dom(p);
(6.3)for all β ∈ Eκω1 ∩ sup(dom(p)) + 1, there is a club C ⊆ β and a 1-1choice function on {p(α) : α ∈ dom(p) ∩ C}.
For p0, p1 ∈ P∗, we deﬁne p1 ≤P∗ p0 if p0 ⊆ p1, dom(p1) is an end extensionof dom(p0) and
(6.4)| dom(p1) \ dom(p0) | > ℵ0 if p0 ̸= p1.
The following lemma can be proved still without using MA+(σ-closed).
Lemma 6.2. (1) For each ξ < κ, the set
Dξ = {p ∈ P∗ : sup(dom(p)) > ξ, dom(p) has the maximal element}
is dense in P∗.
(2)P∗ is σ-closed.
(3)P∗ is κ-strategically closed.
(4)Let ˙SG be a P∗-name of
(6.5)SG = ∪{dom(p) : p ∈ G}
for (V, P∗)-generic ﬁlter G. Then we have ∥– P∗ “ ˙SG ⊆ Eκω ”, ∥– P∗ “ ˙SG is stationary ”and ∥– P∗ “ Eκω \ ˙SG is stationary ”.
(5)∥– P∗ “ ¬FRP(κ) ”.
(6)P∗ collapses all cardinals in the interval (κ, 2<κ].
Proof. (1): Suppose ξ < κ and p ∈ P∗. Let η = max{ξ, sup(dom(p))}. Let qbe any ladder system extending p with
(6.6)dom(q) = dom(p) ∪ {α ∈ Eκω : η < α ≤ η + (ω1 + ω)α = γ + ω for some γ < α}.
Then q ∈ P∗ and thus q ≤P∗ p, and q ∈ Dξ: To see that q satisﬁes thecondition (6.3) for β = η + ω1, let C = {α ∈ Eβω : η < α and α ̸= γ + ω for anyγ ∈ α}. C is then a club below β disjoint from dom(q).
(2): If qn, n ∈ ω is a descending sequence in P∗ then ∪
n∈ω qn is the lowerbound of the sequence.
For (3), Let σ be any strategy of Player II in ⅁κ(P∗) such that σ satisﬁesthe following at the ξ’th inning for ξ < κ.
Case I. ξ is a successor ordinal. Suppose that pξ is the ξ’th move of PlayerI. Then, let qξ, the ξ’th move of Player II to be taken according to σ, be anyq ∈ P∗ such that q ≤P∗ pξ and q ̸= pξ. Note that we can always take such qξ by(1).
Case II. ξ is a limit ordinal. If the set of the previous moves {pη, qη : η < ξ}has a lower bound, then
η<ξ qη ∪ {⟨β, b⟩}
is also a lower bound of this set where β = sup(∪
η<ξ dom(qη)))+ω and b ∈ [β]ℵ0
is a coﬁnal subset of β of order type ω. Let ξ’th move qξ of the player II to betaken according to σ be such q.We show that σ as above is a winning strategy for Player II. Since wealready saw that Player II can take his move at successor steps according toσ, it is enough to show that, for any limit ξ < κ and any (partial) match⟨pη, qη : η < ξ⟩ in ⅁κ(P∗) where Player II has played according to σ, there is alower bound of {pη, qη : η < ξ}. If cf(ξ) = ω then this holds by (2). So supposethat cf(ξ) ̸= ω. Then the set C of all limit points of {sup(dom(qη)) : η < ξ} isa club in β = sup(∪
η<ξ dom(qη)) and C ∩ (∪
η<ξ dom(qη)) = ∅ by the deﬁnitionof qη’s at limit ordinals η < ξ taken according to σ. It follows that q = ∪
η<ξ qηsatisﬁes (6.3). Thus q ∈ P∗ and q is a lower bound of {qη : η < ξ}.
(4): ∥– P∗ “ ˙SG ⊆ Eκω ” follows from (6.1). To show ∥– P∗ “ ˙SG is stationary ”,let ˙C be a P∗-name of a club subset of κ. We have to show that ∥– P∗ “ ˙SG ∩ ˙C ̸=∅ ”. For any p ∈ P∗, let ⟨pn : n ∈ ω⟩ be a decreasing sequence in P∗ and⟨αn : n ∈ ω⟩ an increasing sequence of ordinals below κ such that
(6.7)p0 ≤P∗ p;
(6.8)pn ∥– P∗ “ αn ∈ ˙C ”; and
(6.9)sup(dom(pn+1)) > αn.
Let α = supn∈ω αn. We have α = sup(∪
n∈ω pn) by (6.9). Let
(6.10) q = ∪
n<ω pn ∪ {⟨α, {αn : n ∈ ω}⟩}.
Then q ∈ P∗, q ≤P∗ pn for all n ∈ ω, and hence q ≤P∗ p by (6.7), and q ∥– P∗ “ α ∈˙SG ∩ ˙C ” by (6.10) and (6.8). This shows that ∥– P∗ “ ˙SG ∩ ˙C ̸= ∅ ”. It followsthat ∥– P∗ “ ˙SG is stationary ”.∥– P∗ “ Eκω \ ˙SG is stationary ” can be proved similarly.
(5): Let G be a (V, P∗)-generic ﬁlter. In V [G], SG of (6.5) is stationary by(4). SG ⊆ Eκω by (6.1). gG = ∪ G is a ladder system on SG by (6.2) and thisladder system is a counter example to FRP(κ) by (6.3).
(6): It is enough to show that P∗ adds a surjection from a subset of κ to
κ>2.We work in V [G]. For α ∈ SG, let s(α) be the element of ξ2 where ξ is suchthat the element of SG next to α is α+(ξ +ω) and, for η < ξ, s(α)(η) = 0 if andonly if the ﬁrst element of gG(β) is even where β is the η’th element from belowof SG \ (α + (ξ + ω + 1)). By the genericity of SG and by a slight modiﬁcationof the proofs of (1) and (3), it is easy to see that s : SG → κ>2 is a surjection.
(Lemma 6.2)
We need the following Fact 6.3 for the proof of Theorem 6.1.
Fact 6.3. (B. K¨onig and Y. Yoshinobu [16]) MA+(σ-closed) is preserved bystrongly ω1 + 1-strategically closed forcing.
For the proof of Fact 6.3 the reader may refer to [16].Let G be a (V, P∗)-generic ﬁlter. Working in V [G], let SG be deﬁned as in(6.5). gG = ∪ G is a ladder system on SG.Let Q0 be the following partial ordering: Elements of Q0 are pairs ⟨q, f⟩such that
(6.11) q is a strictly increasing continuous mapping from a successor ordinal< ω1 to κ;
(6.12) rng(q) ⊆ SG;
(6.13) dom(f) = rng(q); and
(6.14) f is an injective choice function on {gG(α) : α ∈ rng(q)}.
For ⟨q0, f0⟩, ⟨q1, f1⟩ ∈ Q0,
(6.15) ⟨q1, f1⟩ ≤Q0 ⟨q0, f0⟩ if q0 ⊆ q1 and f0 ⊆ f1.
Lemma 6.4. (1)For α < ω1 and β < κ, the set
{⟨q, f⟩ ∈ Q0 : dom(q) ≥ α and max(rng(q)) ≥ β}
is dense in Q0.
(2)Q0 adds the canonical strictly and continuously increasing mapping π fromω1 to SG whose image is coﬁnal in κ, as well as the canonical injective choicefunction ϕ on {gG(α) : α ∈ rng(π)}.
(3)For any stationary S ⊆ SG, Q0 forces that S ∩ π ′′ω1 for π as in (the proofof) (2) is stationary.
Proof. (1) is clear and (2) follows immediately from (1) and the deﬁnition ofQ0. More speciﬁcally,
(6.16) π = ∪{q : ⟨q, f⟩ ∈ H for some f} and
ϕ = ∪{f : ⟨q, f⟩ ∈ H for some q}
for a (Q0, V [G])-generic ﬁlter H are the canonical objects.(3): Suppose that ⟨q, f⟩ ∈ Q0 and ˙C is a Q0-name of a club subset of κ.Let θ be a suﬃciently large regular cardinal. By Fact 5.3, there is a countableelementary submodel M of H(θ) such that
(6.17) S, gG, ⟨q, f⟩, ˙C, · · · ∈ M;
(6.18) sup(κ ∩ M) ∈ S; and
(6.19) gG(sup(κ ∩ M)) ̸⊆ κ ∩ M.
Let ⟨γn : n ∈ ω⟩ be an increasing sequence of countable ordinals coﬁnal inω1 ∩ M and ⟨δn : n ∈ ω⟩ be an increasing sequence of ordinals ∈ κ ∩ M coﬁnalin κ ∩ M. Let ⟨qn, fn⟩ ∈ Q0 ∩ M, n ∈ ω be a decreasing sequence such that
(6.20) ⟨q0, f0⟩ ≤Q0 ⟨q, f⟩;
(6.21) ⟨qn, fn⟩ ∥– Q0 “ αn ∈ ˙C ” for some αn ∈ (κ ∩ M) \ δn for all n ∈ ω; and
(6.22) sup(dom(qn)) ≥ γn and sup(rng(qn)) ≥ δn for all n ∈ ω.
By (6.19), there is some
(6.23) α∗ ∈ gG(sup(κ ∩ M)) \ M.
(6.24) q∗ = ∪
n∈ω qn ∪ {⟨sup(ω1 ∩ M), sup(κ ∩ M)⟩} and
(6.25) f ∗ = ∪
n∈ω fn ∪ {⟨sup(κ ∩ M), α∗⟩}.
Note that sup(κ ∩ M) ∈ SG by (6.18) and S ⊆ SG. Hence rng(q∗) ⊆ SG.Since ⟨qn, fn⟩ ∈ M, we have qn ∈ M for each n ∈ ω and
(6.26) fn(β) ∈ M for all n ∈ ω and β ∈ rng(qn).
By (6.22), it follows that
(6.27) sup(∪
n∈ω dom(qn)) = sup(ω1 ∩ M) and
n∈ω rng(qn)) = sup(κ ∩ M).
By (6.23) and (6.26), f ∗ is an injective choice function of {g(β) : β ∈ rng(q∗)}.Thus we have ⟨q∗, f ∗⟩ ∈ Q0 and ⟨q∗, f ∗⟩ ≤Q0 ⟨qn, fn⟩ for n ∈ ω. By (6.20),⟨q∗, f ∗⟩ ≤Q0 ⟨q, f⟩.By (6.27), (6.21), and since˙C is a Q0-name of a clubset ⊆ κ, we have ⟨q∗, f ∗⟩ ∥– Q0 “ sup(κ ∩ M) ∈˙C ”.By (6.18) and (6.24),⟨q∗, f ∗⟩ ∥– Q0 “ sup(κ∩M) ∈ S ∩ ˙π ′′ω1 ”. Thus, ⟨q∗, f ∗⟩ ∥– Q0 “ S ∩ ˙π ′′ω1 ∩ ˙C ̸= ∅ ”where ˙π for a Q0-name of π in (6.16).(Lemma 6.4)
Stepping back to V , let ˙Q0 be a P∗-name of Q0. By Lemma 6.4, we have∥– P∗∗ ˙Q0 “ ˙S ∩ ˙π ′′ω1 is stationary ” for all P∗-name ˙S of stationary subset of ˙SG.
(6.28)D0 = {⟨p, ⟨ ˙q, ˙f⟩⟩ ∈ P∗ ∗ ˙Q0 : p decides ˙q and ˙f to be some q and f
in V and sup(dom(p)) = sup(rng(q))}.
By the deﬁnition of D0, each element ⟨p, ⟨ ˙q, ˙f⟩⟩ of D0 corresponds uniquely to⟨p, ⟨q, f⟩⟩ (with q, f ∈ V ) where p ∥– P∗ “ q = ˙q and f = ˙f ”, and hence
(6.29) q is a strictly and continuously increasing mapping in V with dom(q) ∈ω1 \ Lim(ω1);
(6.30) rng(q) ⊆ dom(p) and sup(rng(q)) = sup(dom(p));
(6.31) dom(f) = rng(q); and
(6.32) f is an injective choice function on {p(α) : α ∈ rng(q)}.
Let D∗0 be the collection of all such ⟨p, ⟨q, f⟩⟩. Then the partial ordering ≤D∗0on D∗0 corresponding to ≤P∗∗ ˙Q0↾ D0 is
(6.33) ⟨p1, ⟨q1, f1⟩⟩ ≤D∗0 ⟨p0, ⟨q0, f0⟩⟩⇔ p1 ≤P∗ p0, q1 ⊇ q0 and f1 ⊇ f0.
Lemma 6.5. (1)D0 is dense in P∗ ∗ ˙Q0.(2)D0 is σ-closed (with respect to ≤P∗∗ ˙Q0↾ D0).
Proof. (1) follows from the κ-strategical closedness of P∗.For (2), it is enough to show that ⟨D∗0, ≤D∗0⟩ is σ-closed.Suppose that S = ⟨⟨pn, ⟨qn, fn⟩⟩ : n < ω⟩ is a strictly decreasing chain in⟨D∗0, ≤D∗0⟩. We have to show that S has a lower bound in D∗0 with respect to ≤D∗0.Let p∗ = ∪
n<ω pn, q∗ = ∪
n<ω qn and f ∗ = ∪
n<ω fn. Let γ∗ = sup(dom(q∗))and δ∗ = sup(dom(p∗)). We have δ∗ = sup(rng(q∗)) = sup(dom(f ∗)). By (6.4)and by the deﬁnition of ≤D∗0, δ∗ \ rng(f ∗) is coﬁnal in δ∗ and cf(δ∗) = ω. Sowe can take a subset c∗ of δ∗ \ rng(f ∗) of order type ω which is coﬁnal in δ∗.Pick an element δ0 of c∗ and, let p† = p∗ ∪ {⟨δ∗, c∗⟩}, q† = q∗ ∪ {⟨γ∗, δ∗⟩} andf † = f ∗∪{⟨δ∗, δ0⟩}. Then we have ⟨p†, ⟨q†, f †⟩⟩ ∈ D∗0 and ⟨p†, ⟨q†, f †⟩⟩ is a lowerbound of S.(Lemma 6.5)
Lemma 6.6 (MA+(σ-closed)). P∗ forces that every stationary subsets of SGreﬂect.
Proof. Let ˙S be a P∗-name of a stationary subset of SG and p0 ∈ P∗. We showthat there are p1 ≤P∗ p0, δ∗ ∈ Eκω1 and S∗ ⊆ δ∗ (in V ) stationary in δ∗ such thatp∗ ∥– P∗ “ S∗ ⊆ ˙S ”.Let us denote by p0 and ˙S also the element of P∗ ∗ ˙Q0 and the P∗ ∗ ˙Q0-namecorresponding to them. Let ˙π be a P∗ ∗ ˙Q0-name of the canonical mapping πfrom ω1 to SG and ˙ϕ a P∗ ∗ ˙Q0-name of the canonical injective choice functionϕ for gG added generically by ˙Q0 as in Lemma 6.4, (2).Let ˙T be the P∗ ∗ ˙Q0-name of ˙π−1 ′′ ˙S. By Lemma 6.4, (3), and since ˙π isforced to be continuous with ˙π ′′ω1 coﬁnal in SG, we have
(6.34)∥– P∗∗ ˙Q0 “ ˙T is a stationary subset of ω1 ”.
By MA+(σ-closed) and by Lemma 6.5, there is a ﬁlter F on P∗ ∗ ˙Q0 such that
(6.35) F is generated from F ∩ D0;
(6.36) ⟨p0, ⟨ˇ∅, ˇ∅⟩⟩ ∈ F;
(6.37)˙T F is a stationary subset of ω1; and
(6.38)˙πF is a strictly and continuously increasing function from ω1 to κ.
(6.39) p∗ = ∪{p : ⟨p, ⟨q, f⟩⟩ ∈ F ∩ D0 for some q and f},
q∗ = ∪{q : ⟨p, ⟨q, f⟩⟩ ∈ F ∩ D0 for some p and f} and
f ∗ = ∪{f : ⟨p, ⟨q, f⟩⟩ ∈ F ∩ D0 for some p and q}.
By (6.37) and (6.38), S∗ = ˙πF ′′ ˙T F is stationary subset of δ∗ = sup(rng(q∗)).By (6.38), cf(δ∗) = ω1. Since F is a ﬁlter, p∗ is a ladder system on dom(p∗) ⊆ Eκωand sup(dom(p∗)) = δ∗. f ∗ witnesses that p∗ satisﬁes (6.3) for β = sup(dom(p)) =δ∗ and hence we have p∗ ∈ P∗. Clearly p∗ ≤P∗ p0 and p∗ ∥– P∗ “ S∗ ⊆ ˙S ”.
(Lemma 6.6)
Now we are going to prove that every stationary subsets of Eκω \ SG reﬂectin V [G]. Working in V [G] again for some (V, P∗)-generic ﬁlter G, let SG and gGbe as before. Let Q1 be the following partial ordering: The elements of Q1 aremappings q such that
(6.40) q is a strictly and continuously increasing mapping from a successorordinal < ω1 to κ; and
(6.41) q ′′Lim(dom(q)) ⊆ Eκω \ SG.
Note that Eκω \ SG is stationary by Lemma 6.2, (4).The following lemma can be proved similarly to Lemma 6.4.
Lemma 6.7. (1)For α < ω1 and β < κ, the set
{q ∈ Q1 : dom(q) ≥ α and max(rng(q)) ≥ β}
is dense in Q1.
(2)Q1 adds the canonical strictly and continuously increasing mapping π1 =∪ H for a (V [G], Q1)-generic ﬁlter H. π1 : ω1 → κ, π1 ′′ω1 is coﬁnal in κ andπ1 ′′Lim(ω1) ⊆ Eκω \ SG.
(3)For any stationary S ⊆ Eκω \ SG, Q1 forces that S ∩ π1 ′′ω1 for π1 as in (2)is stationary.
Stepping back to V , let ˙Q1 be a P∗-name of Q1. By Lemma 6.7, we have∥– P∗∗ ˙Q1 “ ˙S ∩ ˙π1 ′′ω1 is stationary ” for all P∗-name ˙S of stationary subset of Eκω \˙SG and P1-name ˙π1 of π1.
(6.42)D1 = {⟨p, ˙q⟩ ∈ P∗ ∗ ˙Q1 : p decides ˙q to be some q in V
and sup(dom(p)) = sup(rng(q))}.
By the deﬁnition of D1, each element ⟨p, ˙q⟩ of D1 corresponds uniquely to ⟨p, q⟩(with q ∈ V and p ∥– P∗ “ ˙q = q ”) where
(6.43) q is a continuous strictly increasing mapping in V with dom(q) being asuccessor ordinal in ω1;
(6.44) q ′′Lim(dom(q)) ⊆ sup(dom(p)) \ dom(p); and
(6.45) sup(rng(q)) = sup(dom(p)).
Let D∗1 be the collection of all such ⟨p, q⟩. Then the partial ordering ≤D∗1 on D∗1corresponding to ≤P∗∗ ˙Q1↾ D1 is
(6.46) ⟨p1, q1⟩ ≤D∗1 ⟨p1, q1⟩⇔ p1 ≤P∗ p0 and q1 ⊇ q0.
Lemma 6.8. (1)D1 is dense in P∗ ∗ ˙Q1.(2)D1 is σ-closed (with respect to ≤P∗∗ ˙Q1↾ D1).
Proof. We only prove (2) since (1) follows immediately form the κ-strategicalclosedness of P∗.It is enough to show that ⟨D∗1, ≤D∗1⟩ is σ-closed. Suppose that S = ⟨⟨pn, qn⟩ :n < ω⟩ is a strictly decreasing chain in ⟨D∗1, ≤D∗1⟩. We have to show that S hasa lower bound in D∗1 with respect to ≤D∗1. Let p∗ = ∪
n<ω pn and q∗ = ∪
n<ω qn.Let γ∗ = sup(dom(q∗)) and δ∗ = sup(dom(p∗)). Let c∗ = {δ∗ + n : n ∈ ω},p† = p∗ ∪ {⟨δ∗ + ω, c∗⟩} and q† = q∗ ∪ {⟨γ∗, δ∗⟩, ⟨γ∗ + 1, δ∗ + ω⟩}. Then we have⟨p†, q†⟩ ∈ D∗1 and ⟨p†, q†⟩ is a lower bound of S.(Lemma 6.8)
By an argument parallel to the proof of Lemma 6.6 we can now prove thefollowing:
Lemma 6.9 (MA+(σ-closed)). P∗ forces that every stationary subsets of Eκω\SGreﬂect.
Now we show that P∗ forces FRP(λ) for all regular cardinals > 2<κ. Weconsider again a two step iteration over P∗.Working again in V [G] for a (V, P∗)-generic ﬁlter G, let Q2 be the standardforcing for shooting a club through κ \ SG by conditions of size < κ: Theelements q of Q2 are closed bounded subsets of κ such that Lim(q) ⊆ κ \ SG(with the maximal element) and, for q0, q1 ∈ Q2, q1 ≤Q2 q0 if q1 is an endextension of q0.In V , let ˙Q2 be a P∗-name of Q2 and let
(6.47)D2 = {⟨p, ˙q⟩ ∈ P∗ ∗ ˙Q2 : p decides ˙q to be some q in V
and sup(dom(p)) = sup(rng(q))} and
D∗2 = {⟨p, q⟩ : there is a ⟨p, ˙q⟩ ∈ D2 such thatp decides ˙q to be q}
D∗2 with the ordering deﬁned by
(6.48) ⟨p1, q1⟩ ≤D∗2 ⟨p0, q0⟩ if p1 ≤P∗ p0 and q1 is an end extension of q0
for ⟨p0, q0⟩, ⟨p1, q1⟩ ∈ D∗2 is isomorphic to ⟨D2, ≤P∗∗ ˙Q2↾ D2⟩ modulo possiblemultitude of P∗-names for q in ⟨p, q⟩ ∈ D∗2.We can prove the following similarly to the corresponding lemmas for ˙Q0and ˙Q1:
Lemma 6.10. (1)D2 is dense in P∗ ∗ ˙Q2.(2)D2 is κ-closed.
Lemma 6.11 (MA+(σ-closed)). P∗ forces FRP(λ) for all regular λ > 2<κ.
Proof. By | D∗2 | = 2<κ and by Lemma 6.10, (1), P∗ ∗ ˙Q2 has the (2<κ)+-c.c.Hence all (regular) cardinals > 2<κ are preserved.By Lemma 6.10, (1) and (2), P∗ ∗ ˙Q2 is strongly < κ-strategically closed.Hence, by Fact 6.3, we have ∥– P∗∗ ˙Q2 “ MA+(σ-closed) ”. In particular, we have∥– P∗∗ ˙Q2 “ FRP(λ) ”.Suppose that p0 ∈ P∗, ˙S is a P∗-name of a stationary subset of Eλω and ˙ga P∗-name for a ladder system on ˙S. Since ∥– P∗ “ | ˙Q2 | = 2<κ ”, ˙S seen as aP∗ ∗ ˙Q2-name is forced to be a stationary subset of Eλω. By FRP(λ) in V P∗∗ ˙Q2,there is a P∗ ∗ ˙Q2-name ⟨ ˙Iα : α < ω1⟩ of a ﬁltration of a subset ˙I of λ ofcardinality ℵ1 such that P∗ ∗ ˙Q2 forces
(6.49) cf( ˙I) = ω1;
(6.50)˙I is closed with respect to ˙g; and
(6.51) {α ∈ ω1 : sup( ˙Iα) ∈ ˙S and ˙g(sup( ˙Iα)) ⊆ Iα} is stationary.
By Lemma 6.10, we have ∥– P∗ “ ˙Q2 is < κ-Baire ”. Hence ⟨Iα : α < ω1⟩ ∈ V P∗
and this sequence witnesses FRP(λ) for S and g in V P∗.(Lemma 6.11)
We can now prove Theorem 6.1 just by recapitulating all the results weobtained so far.
Proof of Theorem 6.1:We show that the partial ordering P∗ deﬁned in(6.1) ∼ (6.4) has the desired property (1) ∼ (3).(1): We have ∥– P∗ “ ¬FRP(κ) ” by Lemma 6.2, (5). To show ∥– P∗ “ ORP(κ) ”,suppose that p ∈ P∗ and ˙S is a P∗-name of a stationary subset of Eκω. Thenthere is p′ ≤P∗ p, such that either p′ ∥– P∗ “ ˙S ∩ ˙SG is stationary ” or p′ ∥– P∗ “ ˙S \˙SG is stationary ” holds in either case we can conclude that p′ forces that ˙Sreﬂects by Lemma 6.6 or Lemma 6.9.
(2): Since P∗ is κ-strategically closed by Lemma 6.2, (3), FRP(λ) is preservedfor all regular λ with ℵ1 < λ < κ. For regular λ > 2<κ, we have ∥– P∗ “ FRP(λ) ”by Lemma 6.11.
(3): For regular ℵ2 ≤ λ < κ, we have ∥– P∗ “ FRP(λ) ” by MA+(σ-closed)and κ strategically closedness of P∗. For λ > κ forced to be a regular cardinalby P∗, Lemma 6.2, (6) and (2) above imply ∥– P∗ “ FRP(λ) ” and this togetherwith (1) implies ∥– P∗ “ ORP ”.(Theorem 6.1)
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Authors’ addresses
Saka´e Fuchino
Kobe UniversityE-mail address: fuchino@diamond.kobe-u.ac.jp
Lajos Soukup
Alfr´ed R´enyi Institute of MathematicsE-mail address: soukup@renyi.hu
Hiroshi Sakai
Kobe UniversityE-mail address: hsakai@kurt.scitec.kobe-u.ac.jp
Toshimichi Usuba
Nagoya UniversityE-mail address: usuba@math.cm.is.nagoya-u.ac.jp