Set systems without a simplex, Helly hypergraphs and union-efficient families

Stijn Cambie; Nika Salia

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arXiv · 2210.16211

Set systems without a simplex, Helly hypergraphs and union-efficient families

Stijn Cambie·Nika Salia

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Abstract

We present equivalent formulations for concepts related to set families for which every subfamily with empty intersection has a bounded sub-collection with empty intersection. Hereby, we summarize the progress on the related questions about the maximum size of such families. In this work we solve a boundary case of a problem of Tuza for non-trivial $q$-Helly families, by applying Karamata's inequality and determining the minimum size of a $2$-self-centered graph for which the common neighborhood of every pair of vertices contains a clique of size $q-2$.

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Stijn Cambie, Nika Salia. 2022-10-28. Set systems without a simplex, Helly hypergraphs and union-efficient families. https://arxiv.org/abs/2210.16211

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