On Berge-Ramsey Problems
Abstract
Given a graph G, a hypergraph H is a Berge copy of F if V(G)⊂V(H) and there is a bijection f:E(G)→E(H) such that for any edge e of G we have e⊂f(e). We study Ramsey problems for Berge copies of graphs, i.e. the smallest number of vertices of a complete r-uniform hypergraph, such that if we color the hyperedges with c colors, there is a monochromatic Berge copy of G.
We obtain a couple results regarding these problems. In particular, we determine for which r and c the Ramsey number can be super-linear. We also show a new way to obtain lower bounds, and improve the general lower bounds by a large margin. In the specific case G=Kn and r=2c−1, we obtain an upper bound that is sharp besides a constant term, improving earlier results.