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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.

We prove that the maximum number of triangles in a C 5‐free graph on n vertices is at most 122(1+o(1))n3∕2, improving an estimate of Alon and Shikhelman.

Hakimi and Schmeichel considered the problem of maximizing the number of cycles of a given length in an n-vertex planar graph. They determined this number exactly for triangles and 4cycles and conjectured the solution to the problem for 5-cycles. We confirm their conjecture.

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