New Results on Simplex-Clusters in Set Systems

Currier, Gabriel

doi:10.1007/s00493-020-4441-1

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2021

2022

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Cited by 3 publications

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“…Therefore, a natural generalization of Theorem 1.1 would be to maximize the number of

(r+1)

‐triangles in r ‐wise intersecting families. Let us remark that r ‐triangles have been studied under the name of r ‐simplices or r ‐clusters , see, for example, [3, 13–15].…”

Section: Introductionmentioning

confidence: 99%

Triangles in intersecting families

Nagy

Patkós

2022

Mathematika

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“…Therefore, a natural generalization of Theorem 1.1 would be to maximize the number of

(r+1)

‐triangles in r ‐wise intersecting families. Let us remark that r ‐triangles have been studied under the name of r ‐simplices or r ‐clusters , see, for example, [3, 13–15].…”

Section: Introductionmentioning

confidence: 99%

Triangles in intersecting families

Nagy

Patkós

2022

Mathematika

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On the d-cluster generalization of Erdős-Ko-Rado

Currier

2021

Journal of Combinatorial Theory, Series A

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Hypergraphs without non-trivial intersecting subgraphs

Liu

2022

Combinator. Probab. Comp.

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A hypergraph $\mathcal{F}$ is non-trivial intersecting if every pair of edges in it have a nonempty intersection, but no vertex is contained in all edges of $\mathcal{F}$ . Mubayi and Verstraëte showed that for every $k \ge d+1 \ge 3$ and $n \ge (d+1)k/d$ every $k$ -graph $\mathcal{H}$ on $n$ vertices without a non-trivial intersecting subgraph of size $d+1$ contains at most $\binom{n-1}{k-1}$ edges. They conjectured that the same conclusion holds for all $d \ge k \ge 4$ and sufficiently large $n$ . We confirm their conjecture by proving a stronger statement.They also conjectured that for $m \ge 4$ and sufficiently large $n$ the maximum size of a $3$ -graph on $n$ vertices without a non-trivial intersecting subgraph of size $3m+1$ is achieved by certain Steiner triple systems. We give a construction with more edges showing that their conjecture is not true in general.

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New Results on Simplex-Clusters in Set Systems

Cited by 3 publications

References 15 publications

Triangles in intersecting families

Triangles in intersecting families

On the d-cluster generalization of Erdős-Ko-Rado

Hypergraphs without non-trivial intersecting subgraphs

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