Regular subgraphs of uniform hypergraphs

TL;DR

This paper establishes bounds on the maximum edges in large uniform hypergraphs that do not contain r-regular subgraphs, revealing exact extremal structures under certain conditions.

Contribution

It provides new extremal bounds for r-regular subgraph-free hypergraphs and characterizes the extremal configurations for specific parameters.

Findings

Maximum edges are approximately {{n-1}race{k-1}} for large hypergraphs without r-regular subgraphs.

Exact maximum is achieved by hypergraphs where all edges contain a specific vertex for certain r and k.

The paper raises related open questions about regular subgraphs in hypergraphs.

Abstract

We prove that for every integer $r\geq 2$, an $n$-vertex $k$-uniform hypergraph $H$ containing no $r$-regular subgraphs has at most $(1+o(1)){{n-1}\choose{k-1}}$ edges if $k\geq r+1$ 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 ${{n-1} \choose {k-1}}$, with equality only if all edges contain a specific vertex $v$. We also ask some related questions.