Random Turán Problems for Hypergraph Expansions

ABSTRACT Given an ‐uniform hypergraph , we define its ‐uniform expansion to be the hypergraph obtained from by inserting distinct vertices into each edge of , and we define to be the largest ‐free subgraph of the random hypergraph . We initiate the first systematic study of for general hypergraphs . Our main result essentially resolves this problem for large by showing that goes through three predictable phases whenever is Sidorenko and is sufficiently large, with the behavior of being provably more complex whenever has no Sidorenko expansion. Moreover, our methods unify and generalize almost all previously known results for the random Turán problem for degenerate hypergraphs of uniformity at least 3.