Ramsey numbers of Berge-hypergraphs and related structures

For a graph G = (V, E), a hypergraph H is called a Berge-G, denoted by BG, if there exists an injection f: E(G) → E(H) such that for every e ∈ E(G), e ⊆ f(e). Let the Ramsey number Rr(BG, BG) be the smallest integer n such that for any 2-edge-coloring of a complete r-uniform hypergraph on n vertices, there is a monochromatic Berge-G subhypergraph. In this paper, we show that the 2-color Ramsey number of Berge cliques is linear. In particular, we show that R³(BKs, BKt) = s + t - 3 for s, t ≥ 4 and max(s, t) ≥ 5 where BKn is a Berge-Kn hypergraph. We also investigate the Ramsey number of trace hypergraphs, suspension hypergraphs and expansion hypergraphs.