On the cover Ramsey number of Berge hypergraphs

Abstract

For a fixed set of positive integers R, we say H is an R-uniform hypergraph, or R-graph, if the cardinality of each edge belongs to R. An R-graph H is covering if every vertex pair of H is contained in some hyperedge. For a graph G=(V,E), a hypergraph H is called a Berge-G, denoted by BG, if there is an injection i:V(G)→V(H) and a bijection f:E(G)→E(H) such that for all e=uv∈E(G), we have {i(u),i(v)}⊆f(e). In this note, we define a new type of Ramsey number, namely the cover Ramsey number, denoted as RˆR(BG1,BG2), as the smallest integer n0 such that for every covering R-uniform hypergraph H on n≥n0 vertices and every 2-edge-coloring (blue and red) of H , there is either a blue Berge-G1 or a red Berge-G2 subhypergraph. We show that for every k≥2, there exists some ck such that for any finite graphs G1 and G2, R(G1,G2)≤Rˆ[k](BG1,BG2)≤ck⋅R(G1,G2)3. Moreover, we show that for each positive integer d and k, there exists a constant C=C(d,k) such that if G is a graph on n vertices with maximum degree at most d, then Rˆ[k](BG,BG)≤Cn.