A note on Ramsey numbers for Berge-[formula omitted] hypergraphs

Abstract

For a graph G=(V,E), a hypergraph H is called Berge-G if there is a bijection ϕ:E(G)→E(H) such that for each e∈E(G), e⊆ϕ(e). The set of all Berge-G hypergraphs is denoted B(G).For integers k≥2, r≥2, and a graph G, let the Ramsey number Rr(B(G),k) be the smallest integer n such that no matter how the edges of a complete r-uniform n-vertex hypergraph are colored with k colors, there is a copy of a monochromatic Berge-G subhypergraph. Furthermore, let R(B(G),k) be the smallest integer n such that no matter how all subsets of an n-element set are colored with k colors, there is a monochromatic copy of a Berge-G hypergraph.We give an upper bound for Rr(B(G),k) in terms of graph Ramsey numbers. In particular, we prove that when G becomes acyclic after removing some vertex, Rr(B(G),k)≤4k|V(G)|+r−2, in contrast with classical multicolor Ramsey numbers.When G is a triangle (or a K4), we find sharper bounds and some exact results and determine some “small” Ramsey numbers: •k∕2−o(k)≤R3(B(K3),k)≤3k∕4+o(k),•For any odd integer t≠3, R(B(K3),2t−1)=t+2,•2ck≤R3(B(K4),k)≤e(1+o(1))(k−1)k!,•R3(B(K3),2)=R3(B(K3),3)=5,R3(B(K3),4)=6,R3(B(K3),5)=7,R3(B(K3),6)=8,R3(B(K3),8)=9,R3(B(K4),2)=6.