Turán Problems for Expanded Hypergraphs

We obtain new results on the Turán number of any bounded degree uniform hypergraph obtained as the expansion of a hypergraph of bounded uniformity. These are asymptotically sharp over an essentially optimal regime for both the uniformity and the number of edges and solve a number of open problems in Extremal Combinatorics. Firstly, we give general conditions under which the crosscut parameter asymptotically determines the Turán number, thus answering a question of Mubayi and Verstraëte. Secondly, we refine our asymptotic results to obtain several exact results, including proofs of the Huang–Loh–Sudakov conjecture on cross matchings and the Füredi–Jiang–Seiver conjecture on path expansions. We have introduced two major new tools for the proofs of these results. The first of these, Global Hypercontractivity, is used as a ‘black box’ (we present it in a separate paper with several other applications). The second tool, presented in this paper, is a far-reaching extension of the Junta Method, which we develop from a powerful and general technique for finding matchings in hypergraphs under certain pseudorandomness conditions.