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Mathematics > Combinatorics

arXiv:2210.16211 (math)
[Submitted on 28 Oct 2022]

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

Authors: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 $q-2$.
Comments: 14 Pages, 2 figures
Subjects: Combinatorics (math.CO)
MSC classes: 05D05, 05C65, 05C35, 05C69
Cite as: arXiv:2210.16211 [math.CO]
  (or arXiv:2210.16211v1 [math.CO] for this version)
  https://doi.org/10.48550/arXiv.2210.16211

Submission history

From: Stijn Cambie [view email]
[v1] Fri, 28 Oct 2022 15:30:05 UTC (36 KB)
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