Previous work on Ramsey multiplicity focused primarily on the multiplicity of complete graphs of constant size. We study the question for larger complete graphs, specifically showing that in every 2-coloring of a complete graph on @math vertices, there are at least @math monochromatic complete subgraphs of size between @math and @math . We also study bounds on the ratio between a maximum and a random red clique in a graph, and the ratio between a maximum and a random monochromatic clique in a graph.
The conjecture was later disproved by counterexamples in 1989 by Thomason @cite_14 , who showed that it does not hold for @math . Subsequently, several others worked on upper bounds for @math for small @math . Soon after Thomason's work Franek and R "o dl @cite_13 also gave some different counterexamples based on Cayley graphs for @math . Then in 1994, Jagger, S t'ov ' c ek and Thomason @cite_9 studied for which subgraphs the Burr-Rosta conjecture holds, and found that it does not hold for any graph with @math as a subgraph, which is consistent with the @math result found by Thomason.
@cite_9 @cite_14 @cite_13
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Abstract Denote by k t ( G ) the number of cliques of order t in the graph G . Let k t (n)= min k t (G) + k t ( G ): |G| = n , where Ḡ denotes the complement of G , and | G | denotes the order of G . Let c t (n) = k t (n) ( n t , and let c t = lim n →∞ c t ( n ). An old conjecture of Erdos (1962), related to Ramsey's theorem, states that c t = 2 1−( t 2 ) . It was shown false by Thomason (1989) for all t ⩾4. We present a class of simply describable Cayley graphs which also show the falsity of Erdos conjecture for t = 4. These graphs were found by a computer search and, although of large orders (2 10 -2 14 ), they are rather simple and highly regular. The smallest upper bound for c 4 obtained by us is 0.976501 x 1 32 , and is given by the graph on the power set of a 10-element set (and, hence, of order 2 10 ) determined by the configuration 1,3,4,7,8,10 . and by the graph on the power set of 11 elements (and, hence, of order 2 11 ) determined by the configuration 1,3,4,7,8,10,11 . It is also shown that the ratio of edges to nonedges in a sequence contradicting the conjecture for t = 4 may approach 1, unlike in the sequences of graphs Thomason used in 1989. Friedland's Lower Matching Conjecture asserts that if @math is a @math --regular bipartite graph on @math vertices, and @math denotes the number of matchings of size @math , then @math where @math . When @math , this conjecture reduces to a theorem of Schrijver which says that a @math --regular bipartite graph on @math vertices has at least @math perfect matchings. L. Gurvits proved an asymptotic version of the Lower Matching Conjecture, namely he proved that @math In this paper, we prove the Lower Matching Conjecture. In fact, we will prove a slightly stronger statement which gives an extra @math factor compared to the conjecture if @math is separated away from @math and @math , and is tight up to a constant factor if @math is separated away from @math . We will also give a new proof of Gurvits's and Schrijver's theorems, and we extend these theorems to @math --biregular bipartite graphs. This paper considers the problem of designing maximum distance separable (MDS) codes over small fields with constraints on the support of their generator matrices. For any given @math binary matrix @math , the GM-MDS conjecture, proposed by , states that if @math satisfies the so-called MDS condition, then for any field @math of size @math , there exists an @math MDS code whose generator matrix @math , with entries in @math , fits the matrix @math (i.e., @math is the support matrix of @math ). Despite all the attempts by the coding theory community, this conjecture remains still open in general. It was shown, independently by and , that the GM-MDS conjecture holds if the following conjecture, referred to as the TM-MDS conjecture, holds: if @math satisfies the MDS condition, then the determinant of a transform matrix @math , such that @math fits @math , is not identically zero, where @math is a Vandermonde matrix with distinct parameters. In this work, we first reformulate the TM-MDS conjecture in terms of the Wronskian determinant, and then present an algebraic-combinatorial approach based on polynomial-degree reduction for proving this conjecture. Our proof technique's strength is based primarily on reducing inherent combinatorics in the proof. We demonstrate the strength of our technique by proving the TM-MDS conjecture for the cases where the number of rows ( @math ) of @math is upper bounded by @math . For this class of special cases of @math where the only additional constraint is on @math , only cases with @math were previously proven theoretically, and the previously used proof techniques are not applicable to cases with @math . In the Cayley graph of the mapping class group of a closed surface, with respect to any generating set, we look at a ball of large radius centered on the identity vertex, and at the proportion among the vertices in this ball representing pseudo-Anosov elements. A well-known conjecture states that this proportion should tend to one as the radius tends to infinity. We prove that it stays bounded away from zero. We also prove similar results for a large class of subgroups of the mapping class group.