Forbidden Berge Hypergraphs

Anstee, Richard P.; Salazar, Santiago

doi:10.48550/arxiv.1608.03632

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2019

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“…The general definition of a Berge-G for an arbitrary graph G was introduced by Gerbner and Palmer in [10]. Since their publication, the Turán problem for Berge-G-free hypergraphs has been investigated heavily (see, for example [2,22,11]). Complete graphs were considered in [17] (and recently [12]).…”

Section: Introductionmentioning

confidence: 99%

Ramsey Numbers of Berge-Hypergraphs and Related Structures

Salia¹,

Tompkins²,

Wang³

et al. 2019

Electron. J. Combin.

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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.

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