Papers › Ramsey numbers of Berge-hypergraphs and related structures

Ramsey numbers of Berge-hypergraphs and related structures

29 Aug 2018arXiv:1808.09863links table onlyarchive 2025-07-28

Nika Salia, Casey Tompkins, Zhiyu Wang, Oscar Zamora

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For a graph G=(V,E), a hypergraph ℋ is called a Berge-G, denoted by BG, if there exists a bijection 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. For higher uniformity, we show that R⁴(BKₜ, BKₜ) = t+1 for t≥6 and Rᵏ(BKₜ, BKₜ)=t for k ≥5 and t sufficiently large. We also investigate the Ramsey number of trace hypergraphs, suspension hypergraphs and expansion hypergraphs.

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