Regular Subgraphs of Linear Hypergraphs

We prove that the maximum number of edges in a 3-uniform linear hypergraph on vertices containing no 2-regular subhypergraph is n^1+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 vertices containing no immersion of a closed surface is n^2+o(1). Furthermore, we present results on the maximum number of edges in -uniform linear hypergraphs containing no r-regular subhypergraph.