Jozef Skokan

Regularity lemma for k-uniform hypergraphs

Szemeredis Regularity Lemma proved to be a very powerful tool in extremal graph theory with a large number of applications. Chung [Regularity lemmas for hypergraphs and quasi-randomness, Random Structures Algorithms 2 (1991), 241-252], Frankl and Rodl [The uniformity lemma for hypergraphs, Graphs Combin 8 (1992), 309-312; Extremal problems on set systems, Random Structures Algorithms 20 (2002), 131-164] considered several extensions of Szemeredis Regularity Lemma to hypergraphs. In particular, [Extremal problems on set systems, Random Structures Algorithms 20 (2002), 131-164] contains a regularity lemma for 3-uniform hypergraphs that was applied to a number of problems. In this paper, we present a generalization of this regularity lemma to k-uniform hypergraphs. Similar results were recently independently and alternatively obtained by W. T. Gowers.

Hypergraphs, Quasi-randomness, and Conditions for Regularity

Haviland and Thomason and Chung and Graham were the first to investigate systematically some properties of quasi-random hypergraphs. In particular, in a series of articles, Chung and Graham considered several quite disparate properties of random-like hypergraphs of density 1/2 and proved that they are in fact equivalent. The central concept in their work turned out to be the so called deviation of a hypergraph. They proved that having small deviation is equivalent to a variety of other properties that describe quasi-randomness. In this paper, we consider the concept of discrepancy for k-uniform hypergraphs with an arbitrary constant density d (0

The Ramsey number for hypergraph cycles I

Let Cn denote the 3-uniform hypergraph loose cycle, that is the hypergraph with vertices v1.....,vn and edges v1v2v3, v3v4v5, v5v6v7,.....,vn-1vnv1. We prove that every red-blue colouring of the edges of the complete 3-uniform hypergraph with N vertices contains a monochromatic copy of Cn, where N is asymptotically equal to 5n/4. Moreover this result is (asymptotically) best possible.

RAMSEY-GOODNESS—AND OTHERWISE

A celebrated result of Chvátal, Rödl, Szemerédi and Trotter states (in slightly weakened form) that, for every natural number Δ, there is a constant rΔ such that, for any connected n-vertex graph G with maximum degree Δ, the Ramsey number R(G,G) is at most rΔn, provided n is sufficiently large.In 1987, Burr made a strong conjecture implying that one may take rΔ = Δ. However, Graham, Rödl and Ruciński showed, by taking G to be a suitable expander graph, that necessarily rΔ > 2cΔ for some constant c>0. We show that the use of expanders is essential: if we impose the additional restriction that the bandwidth of G be at most some function β(n)=o(n), then R(G,G)≤(2χ(G)+4)n≤(2Δ+6)n, i.e., rΔ=2Δ+6 suffices. On the other hand, we show that Burr’s conjecture itself fails even for Pnk, the kth power of a path Pn.Brandt showed that for any c, if Δ is sufficiently large, there are connected n-vertex graphs G with Δ(G)≤Δ but R(G,K3) > cn. We show that, given Δ and H, there are β>0 and n0 such that, if G is a connected graph on n≥n0 vertices with maximum degree at most Δ and bandwidth at most βn, then we have R(G,H)=(χ(H)−1)(n−1)+σ(H), where σ(H) is the smallest size of any part in any χ(H)-partition of H. We also show that the same conclusion holds without any restriction on the maximum degree of G if the bandwidth of G is at most ɛ(H) log n=log logn.

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 Rodl.

Bipartite Subgraphs and Quasi-Randomness

Abstract.We say that a family of graphs is p-quasi-random, 0<p<1, if it shares typical properties of the random graph G(n,p); for a definition, see below. We denote by the class of all graphs H for which and the number of not necessarily induced labeled copies of H in Gn is at most (1+o(1))pe(H)nv(H) imply that is p-quasi-random. In this note, we show that all complete bipartite graphs Ka,b, a,b≥2, belong to for all 0<p<1.

On the multi-colored Ramsey numbers of cycles

For a graph L and an integer k≥2, Rk(L) denotes the smallest integer N for which for any edge-coloring of the complete graph KN by k colors there exists a color i for which the corresponding color class contains L as a subgraph. Bondy and Erdos conjectured that, for an odd cycle Cn on n vertices, They proved the case when k = 2 and also provided an upper bound Rk(Cn)≤(k+ 2)!n. Recently, this conjecture has been verified for k = 3 if n is large. In this note, we prove that for every integer k≥4, When n is even, Sun Yongqi, Yang Yuansheng, Xu Feng, and Li Bingxi gave a construction, showing that Rk(Cn)≥(k−1)n−2k+ 4. Here we prove that if n is even, then © 2011 Wiley Periodicals, Inc. J Graph Theory 69: 169-175, 2012

Counting Small Cliques in 3-uniform Hypergraphs

Many applications of Szemeredis Regularity Lemma for graphs are based on the following counting result. If

On The Triangle Removal Lemma For Subgraphs of Sparse Pseudorandom Graphs

{\mathcal G}

The Ramsey number of the clique and the hypercube

is an