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 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 R 3 (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 R 4 (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.

Let f (n, H) denote the maximum number of copies of H possible in an n-vertex planar graph. The function f (n, H) has been determined when H is a cycle of length 3 or 4 by Hakimi and Schmeichel and when H is a complete bipartite graph with smaller part of size 1 or 2 by Alon and Caro. We determine f (n, H) exactly in the case when H is a path of length 3.

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The maximum size of an r-uniform hypergraph without a Berge cycle of length at least k has been determined for all k ≥ r + 3 by Füredi, Kostochka and Luo and for k < r (and k = r, asymptotically) by Kostochka and Luo. In this paper, we settle the remaining cases: k = r + 1 and k = r + 2, proving a conjecture of Füredi, Kostochka and Luo.Recently, Füredi, Kostochka and Luo [3] proved exact bounds similar to Theorem 1 for hypergraphs avoiding long Berge cycles.