The Ramsey Number for 3-Uniform Tight Hypergraph Cycles

Let C(3)n denote the 3-uniform tight cycle, that is, the hypergraph with vertices v1, .–.–., vn and edges v1v2v3, v2v3v4, .–.–., vn−1vnv1, vnv1v2. We prove that the smallest integer N = N(n) for which every red–blue colouring of the edges of the complete 3-uniform hypergraph with N vertices contains a monochromatic copy of C(3)n is asymptotically equal to 4n/3 if n is divisible by 3, and 2n otherwise. The proof uses the regularity lemma for hypergraphs of Frankl and Rödl

An optimal algorithm for checking regularity (extended abstract)

We present a deterministic algorithm A that, in 0(m2) time, verifies whether a given m by m bipartite graph G is regular, in the sense of Szemeredi [18]. In the case in which G is not regular enough, our algorithm outputs a vntness to this irregularity. Algorithm A may be used as a subroutine in an algorithm that finds an e-regular partition of a given n-vertex graph Λ in time 0(n2). This time complexity is optimal, up to a constant factor, and improves upon the bound 0(M(n)), proved by Alon, Duke, Lefmann, Rodl, and Yuster [1, 2], where M(n) = 0(n2-376) is the time required to square a 6-1 matrix over the integers. Our approach is elementary, except that it makes use of linear-sized expanders to accomplish a suitable form of deterministic sampling

Geometrical realization of set systems and probabilistic communication complexity

Let d = d(n) be the minimum d such that for every sequence of n subsets F I, F 2,.•., F n of {I, 2,..., n} there exist n points PI ' P 2,..., P n and n hyperplanes HI ' H 2,..., Hn in R d such that P j lies in the positive side of Hi iff j E Fi. Then In this paper we prove: Theorem 1 1. If n,m-+-00 and log2m = o(n) then d(n,m)s(t + o(l))n Put d = d(n, 1n) the

The hypergraph regularity method and its applications

Szemerédi's regularity lemma asserts that every graph can be decomposed into relatively few random-like subgraphs. This random-like behavior enables one to find and enumerate subgraphs of a given isomorphism type, yielding the so-called counting lemma for graphs. The combined application of these two lemmas is known as the regularity method for graphs and has proved useful in graph theory, combinatorial geometry, combinatorial number theory, and theoretical computer science. Here, we report on recent advances in the regularity method for k-uniform hypergraphs, for arbitrary k ≥ 2. This method, purely combinatorial in nature, gives alternative proofs of density theorems originally due to E. Szemerédi, H. Furstenberg, and Y. Katznelson. Further results in extremal combinatorics also have been obtained with this approach. The two main components of the regularity method for k-uniform hypergraphs, the regularity lemma and the counting lemma, have been obtained recently: Rödl and Skokan (based on earlier work of Frankl and Rödl) generalized Szemerédi's regularity lemma to k-uniform hypergraphs, and Nagle, Rödl, and Schacht succeeded in proving a counting lemma accompanying the Rödl–Skokan hypergraph regularity lemma. The counting lemma is proved by reducing the counting problem to a simpler one previously investigated by Kohayakawa, Rödl, and Skokan. Similar results were obtained independently by W. T. Gowers, following a different approach

The Ramsey number for hypergraph cycles II

Let C(3)n denote the 3-uniform tight cycle, that is the hypergraph with vertices v1, . . . , vn and edges v1v2v3, v2v3v4, . . . , vn-1vnv1, vnv1v2. We prove that the smallest integer N = N(n) for which every red-blue coloring of the edges of the complete 3-uniform hypergraph with N vertices contains a monochromatic copy of C(3)n is asymptotically equal to 4n/3 if n is divisible by 3, and 2n otherwise. The proof uses the regularity lemma for hypergraphs of Frankl and Rödl

Universality, Tolerance, Chaos and Order

What is the minimum possible number of edges in a graph that contains a copy of every graph on n vertices with maximum degree a most k? This question, as well as several related variants, received a considerable amount of attention during the last decade. In this short survey we describe the known results focusing on the main ideas in the proofs, discuss the remaining open problems, and mention a recent application in the investigation of the complexity of subgraph containment problems