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arXiv · 2005.04282

Maximum size intersecting families of bounded minimum positive co-degree

József Balogh·Nathan Lemons·Cory Palmer
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Abstract

Let $\mathcal{H}$ be an $r$-uniform hypergraph. The \emph{minimum positive co-degree} of $\mathcal{H}$, denoted by $\delta_{r-1}^+(\mathcal{H})$, is the minimum $k$ such that if $S$ is an $(r-1)$-set contained in a hyperedge of $\mathcal{H}$, then $S$ is contained in at least $k$ hyperedges of $\mathcal{H}$. For $r\geq k$ fixed and $n$ sufficiently large, we determine the maximum possible size of an intersecting $r$-uniform $n$-vertex hypergraph with minimum positive co-degree $\delta_{r-1}^+(\mathcal{H}) \geq k$ and characterize the unique hypergraph attaining this maximum. This generalizes the Erd\H os-Ko-Rado theorem which corresponds to the case $k=1$. Our proof is based on the delta-system method.

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József Balogh, Nathan Lemons, Cory Palmer. 2020-05-08. Maximum size intersecting families of bounded minimum positive co-degree. https://arxiv.org/abs/2005.04282

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