Regular subgraphs of uniform hypergraphs

Abstract

We prove that for every integer $r\ge 2$, an $n$-vertex $k$-uniform hypergraph $H$ containing no $r$-regular subgraphs has at most $(1+o(1))\left(\genfrac{}{}{0ex}{}{n-1}{k-1}\right)$ edges if $k\ge r+1$ and $n$ is sufficiently large. Moreover, if $r\in \{3,4\}$, $r\mid k$ and $k,n$ are both sufficiently large, then the maximum number of edges in an $n$-vertex $k$-uniform hypergraph containing no $r$-regular subgraphs is exactly $\left(\genfrac{}{}{0ex}{}{n-1}{k-1}\right)$, with equality only if all edges contain a specific vertex $v$. We also ask some related questions.

Cited by in corpus (1)

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