Abstract: For a k-uniform hypergraph G, the Ramsey number R(G,G) is the least N such that in every 2-coloring of edges of a complete n-vertex k-uniform hypergraph, there is a monochromatic copy of G. A family F of k-uniform hypergraphs is f(n)-Ramsey if there is a positive constant c such that R(G,G) <= c f(|V(G)|) for every G\in F.
Burr and Erdös conjectured that for every d, the families M(d) of graphs with maximum degree d and D(d) of d-degenerate graphs are n-Ramsey. Recall that a graph is d-degenerate if each subgraph has a vertex of degree at most d. Chvátal, Rödl, Szemerédi and Trotter proved the first conjecture. The second is open. However, Kostochka and Rödl proved that D(d) is n2-Ramsey, and then Kostochka and Sudakov proved that the family is n1+\epsilon-Ramsey for every positive \epsilon.
In this talk, we prove that for every \epsilon > 0 and every k and d, the family D(d,k) of k-uniform hypergraphs with maximum degree at most d is n1+\epsilon-Ramsey. We also give some examples showing that the graph and hypergraph cases behave differently. This is joint work with Vojtech Rödl.