Vojtěch Rödl

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.

A Dirac-Type Theorem for 3-Uniform Hypergraphs

A Hamiltonian cycle in a 3-uniform hypergraph is a cyclic ordering of the vertices in which every three consecutive vertices form an edge. In this paper we prove an approximate and asymptotic version of an analogue of Diracs celebrated theorem for graphs: for each γ>0 there exists n0 such that every 3-uniform hypergraph on

Threshold functions for Ramsey properties

n\geq n_0

On K 4 -Free Subgraphs of Random Graphs.

vertices, in which each pair of vertices belongs to at least

An approximate Dirac-type theorem for k-uniform hypergraphs

(1/2+\gamma)n

A short proof of the existence of highly chromatic hypergraphs without short cycles

edges, contains a Hamiltonian cycle.

Mathematics of Ramsey theory

Probabilistic methods have been used to approach many problems of Ramsey theory. In this paper we study Ramsey type questions from the point of view of random structures. Let K(n, N) be the random graph chosen uniformly from among all graphs with n vertices and N edges. For a fixed graph G and an integer r we address the question what is the minimum N = N (G, r, n) such that the random graph K(n, N) contains, almost surely, a monochromatic copy of G in every r-coloring of its edges ( K(n, N) -+ (G), , in short). We find a graph parameter )I = )Ie G) yielding { 0 if N < cn1 , llli.~ Prob(K(n, N) -+ (G),) = I if N > Cn1 , for some c, C > o. We use this to derive a number of consequences that deal with the existence of sparse Ramsey graphs. For example we show that for all r ~ 2 and k ~ 3 there exists C > 0 such that almost all graphs H with n vertices and cntfi edges which are Kk+l-free, satisfy H -+ (Kk),. We also apply our method to the problem of finding the smallest N = N(k, r, n) guaranteeing thatalmost all sequences I

Hypergraphs, Quasi-randomness, and Conditions for Regularity

a l < a2 < ... < aN

On the arc-chromatic number of a digraph

n contain an arithmetic progression of length k in every r-coloring, and show k-2 that N = 6( n k=T) is the threshold. DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE, EMORY UNIVERSITY, ATLANTA, GEORGIA 30322 E-mail address: rodlOmathcs. emory. edu E-mail address:rucinskiClvm.amu.edu.pl License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

Generalizations of the removal lemma

For 0<γ≤1 and graphsG andH, writeG→γH if any γ-proportion of the edges ofG spans at least one copy ofH inG. As customary, writeKr for the complete graph onr vertices. We show that for every fixed real η>0 there exists a constantC=C(η) such that almost every random graphGn,p withp=p(n)≥Cn−2/5 satisfiesGn,p→2/3+ηK4. The proof makes use of a variant of Szemerédis regularity lemma for sparse graphs and is based on a certain superexponential estimate for the number of pseudo-random tripartite graphs whose triangles are not too well distributed. Related results and a general conjecture concerningH-free subgraphs of random graphs in the spirit of the Erdős-Stone theorem are discussed.