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 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). Let the Ramsey number R^r (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 (BK_s, BK_t) = s + t − 3 for s, t ≥ 4 and max{s, t} ≥ 5 where BK_n is a Berge-K_n hypergraph. For higher uniformity, we show that R⁴ (BK_t, BK_t) = t + 1 for t ≥ 6 and R^k (BK_t, BK_t) = t for k ≥ 5 and t sufficiently large. We also investigate the Ramsey number of trace hypergraphs, suspension hypergraphs and expansion hypergraphs.