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 R r (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 R r (B (G) , k) in terms of graph Ramsey numbers. In particular, we prove that when G becomes acyclic after removing some vertex, R r (B (G) , k) ≤ 4 k | V (G) | + r − 2 , in contrast with classical multicolor Ramsey numbers. When G is a triangle (or a K 4), we find sharper bounds and some exact results and determine some "small" Ramsey numbers: • k ∕ 2 − o (k) ≤ R 3 (B (K 3) , k) ≤ 3 k ∕ 4 + o (k) , • For any odd integer t ≠ 3 , R (B (K 3) , 2 t − 1) = t + 2 , • 2 c k ≤ R 3 (B (K 4) , k) ≤ e (1 + o (1)) (k − 1) k ! , • R 3 (B (K 3) , 2) = R 3 (B (K 3) , 3) = 5 , R 3 (B (K 3) , 4) = 6 , R 3 (B (K 3) , 5) = 7 , R 3 (B (K 3) , 6) = 8 , R 3 (B (K 3) , 8) = 9 , R 3 (B (K 4) , 2) = 6. [ABSTRACT FROM AUTHOR]