Bounding the number of hyperedges in friendship r-hypergraphs

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Abstract

For r≥ 2, an r-uniform hypergraph is called a friendship r-hypergraph if every set R of r vertices has a unique 'friend' - that is, there exists a unique vertex x ∉. R with the property that for each subset A⊆. R of size r- 1, the set A ∪. {. x} is a hyperedge.We show that for r≥. 3, the number of hyperedges in a friendship r-hypergraph is at least r+1r(n-1r-1), and we characterise those hypergraphs which achieve this bound. This generalises a result given by Li and van Rees in the case when r = 3.We also obtain a new upper bound on the number of hyperedges in a friendship r-hypergraph, which improves on a known bound given by Li, van Rees, Seo and Singhi when r = 3.

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Gunderson, K., Morrison, N., & Semeraro, J. (2016). Bounding the number of hyperedges in friendship r-hypergraphs. European Journal of Combinatorics, 51, 125–134. https://doi.org/10.1016/j.ejc.2015.05.002

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