Weak hypergraph regularity and linear hypergraphs

We consider conditions which allow the embedding of linear hypergraphs of fixed size. In particular, we prove that any k-uniform hypergraph H of positive uniform density contains all linear k-uniform hypergraphs of a given size. More precisely, we show that for all integers l >= k >= 2 and every d > 0 there exists Q > 0 for which the following holds: if His a sufficiently large k-uniform hypergraph with the property that the density of H induced on every vertex subset of size on is at least d, then H contains every linear k-uniform hypergraph F with l vertices. The main ingredient in the proof of this result is a counting lemma for linear hypergraphs, which establishes that the straightforward extension of graph epsilon-regularity to hypergraphs suffices for counting linear hypergraphs. We also consider some related problems. (C) 2009 Elsevier Inc. All rights reserved.

The 3-colored Ramsey number of even cycles

Denote by R(L, L, L) the minimum integer N such that any 3-coloring of the edges of the complete graph on N vertices contains a monochromatic copy of a graph L. Bondy and Erdos conjectured that when L is the cycle C(n) on n vertices, R(C(n), C(n), C(n)) = 4n - 3 for every odd n > 3. Luczak proved that if n is odd, then R(C(n), C(n), C(n)) = 4n + o(n), as n -> infinity, and Kohayakawa, Simonovits and Skokan confirmed the Bondy-Erdos conjecture for all sufficiently large values of n. Figaj and Luczak determined an asymptotic result for the `complementary` case where the cycles are even: they showed that for even n, we have R(C(n), C(n), C(n)) = 2n + o(n), as n -> infinity. In this paper, we prove that there exists n I such that for every even n >= n(1), R(C(n), C(n), C(n)) = 2n. (C) 2009 Elsevier Inc. All rights reserved.