Regular subgraphs of linear hypergraphs

TL;DR

This paper establishes upper bounds on the number of edges in certain linear hypergraphs avoiding specific substructures, resolving a conjecture and extending results to hypergraphs excluding surface immersions.

Contribution

It proves a conjecture on the maximum edges in 3-uniform linear hypergraphs without 2-regular subhypergraphs and applies this to hypergraphs avoiding surface immersions.

Findings

Maximum edges in 3-uniform linear hypergraphs without 2-regular subhypergraphs is n^{1+o(1)}

Maximum edges in 3-uniform hypergraphs without surface immersions is n^{2+o(1)}

Results extend to k-uniform hypergraphs avoiding r-regular subhypergraphs

Abstract

We prove that the maximum number of edges in a 3-uniform linear hypergraph on $n$ vertices containing no 2-regular subhypergraph is $n^{1+o(1)}$. This resolves a conjecture of Dellamonica, Haxell, Luczak, Mubayi, Nagle, Person, R\"odl, Schacht and Verstra\"ete. We use this result to show that the maximum number of edges in a $3$-uniform hypergraph on $n$ vertices containing no immersion of a closed surface is $n^{2+o(1)}$. Furthermore, we present results on the maximum number of edges in $k$-uniform linear hypergraphs containing no $r$-regular subhypergraph.