The maximum measure of 3-wise t-intersecting families

DOI10.1016/J.EJC.2023.103703arXiv2112.08685MaRDI QIDQ6385852

Norihide Tokushige

Publication date: 16 December 2021

Abstract: Let $mathcalG$ be a family of subsets of an $n$-element set. The family $mathcalG$ is called $3$-wise $t$-intersecting if the intersection of any three subsets in $mathcalG$ is of size at least $t$. For a real number $pin(0,1)$ we define the measure of the family by the sum of $p^{|G|}(1-p{)}^{n-|G|}$ over all $GinmathcalG$. We prove that if $tgeq15$ and $pleq2/(sqrt4t+9-1)$ then $p^t$ is the maximum measure of $3$-wise $t$-intersecting families, and the bound for $p$ is sharp. We also present the corresponding stability result for shifted families.

Mathematics Subject Classification ID

Exact enumeration problems, generating functions (05A15) Extremal set theory (05D05)

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