120420900
On a problem of K.F. Roth concerning irregularities of point distribution
204985161
On 3-chromatic hypergraphs
Abstract Let F be uniform hypergraph. In the present paper I prove that of | F | 1 3 4 2 then the chromatic number of F is equal to 2. I have a result in the general (not necessarily uniform) case too.
43488341
Uniformity and Irregularity
Introduction. The purpose of this report is to draw attention to some nontrivial connections between the ("continuous") theory of irregularities of distribution and discrete mathematics. The object of the theory of irregularities of distribution is to measure the uniformity (or nonuniformity) of sequences and point sets. For instance: how uniformly can an arbitrary set of N points in the unit cube be distributed relative to a given family of "nice" sets (e.g., boxes with sides parallel to the coordinate axes, rotated boxes, balls, all convex sets). This theory lies on the border of many branches of mathematics (number theory, discrete geometry, combinatorics, etc.) and has very important applications, e.g., in numerical integration. Here we focus, of course, on the combinatorial aspects of the theory. As an illustration, we shall discuss first two problems of a discrete nature which have fascinating connections with this "continuous" theory. The first question is concerned with balanced two-colorings of finite sets in a square (the problem is due to G. Tusnâdy). Let P = { p i , . . . ,PJV} be a distribution of N points in the unit square [0, l ) 2 . Let / : P —• {—1, +1} be a "two-coloring" of P. Let B denote any rectangle in [0, l ) 2 with sides parallel to the coordinate axes (an aligned rectangle, in short). Consider the function
118653330
AN APPLICATION OF LOVASZ LOCAL LEMMA: THERE EXISTS AN INFINITE 01-SEQUENCE CONTAINING NO NEAR IDENTICAL INTERVALS
Publisher Summary 
This chapter describes an application of Lovasz local lemma. A well-known theorem in combinatorics states that one can associate 0 or 1 with each point of a 2n -cycle (n ≥ 2) in such a way that all arcs of length n of the cycle give different 01-sequences. The chapter presents an equivalent formulation of the result. It is found that given arbitrary natural number n, there exists an infinite 01-sequence such that any two identical intervals of length n have distance at least 2n. The box principle yields the sharpness of this result. It is observed that given arbitrary small ɛ > 0, there is an infinite 01-sequence such that any two identical intervals of length n have distance greater than (2−ɛ)n for all n > n(ɛ).
118985900
New results in the theory of irregularities of point distributions
116996140
On H-spaces and infinite loop spaces