Embeddings and Ramsey numbers of sparse κ-uniform hypergraphs

Abstract

Chvátal, Rödl, Szemerédi and Trotter [3] proved that the Ramsey numbers of graphs of bounded maximum degree are linear in their order. In [6, 23] the same result was proved for 3-uniform hypergraphs. Here we extend this result to k-uniform hypergraphs for any integer k ≥ 3. As in the 3-uniform case, the main new tool which we prove and use is an embedding lemma for k-uniform hypergraphs of bounded maximum degree into suitable k-uniform 'quasi-random' hypergraphs.

3-Uniform hypergraphs of bounded degree have linear Ramsey numbers

“…The strategy for the k-uniform case in [3] is similar to the one used here, but several additional problems arise.…”

“…More precisely, they showed that for all ε, Δ, k > 0 there is a constant C such that R(H ) C|H | 1+ε if H is k-uniform and has maximum degree at most Δ. In a sequel [3] to this paper, we generalised Theorem 1 to k-uniform hypergraphs of bounded degree for arbitrary k. This was also done independently by Ishigami [12]. Another related result is that of Haxell et al [9,10], who asymptotically determined the Ramsey numbers of 3-uniform tight and loose cycles.…”

“…Note that if (G , P ) respects the partition of H and P is ( 3 , r)-regular whenever H contains a hyperedge with vertices in X i , X j , X . Thus if P is also graph-regular, H has bounded maximum degree, δ 3 d 3 and |X i | |V i | for all i, then one might hope that this regularity can be used to find an embedding of H in G (where the vertices in X i are represented by vertices in V i ).…”

Ramsey numbers of sparse hypergraphs

“…Since the first proof of the sparse graph Ramsey theorem used Szemerédi's regularity lemma, it was therefore natural to expect that, given the recent advances in developing a hypergraph regularity method [14,25,23], linear bounds might as well be provable for hypergraphs. Such a program was indeed recently pursued by several authors (Cooley, Fountoulakis, Kühn, and Osthus [6,7]; Nagle, Olsen, Rödl, and Schacht [22]; Ishigami [18]), with the result that we now have the following theorem:…”

Regular slices for hypergraphs

“…Indeed, for regular k-complexes we have a Counting Lemma (see [3,5,6,7,8]), which gives the approximate number of copies of any small fixed k-complex within a regular k-complex, as well an Extension Lemma [2], an Embedding Lemma [2] and (under some additional conditions) a Blow-up Lemma [4], each of which functions similarly as in the graph case.…”