Regular Subgraphs of Linear Hypergraphs

We prove that the maximum number of edges in a 3-uniform linear hypergraph on n vertices containing no 2-regular subhypergraph is n¹⁺ᵒ⁽¹⁾. This resolves a conjecture of Dellamonica, Haxell, Łuczak, Mubayi, Nagle, Person, Rödl, Schacht, and Verstraëte. 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 n²⁺ᵒ⁽¹⁾. Furthermore, we present results on the maximum number of edges in k-uniform linear hypergraphs containing no r-regular subhypergraph.