Disperse hypergraphs

Abstract For $\ell \geq 3$ , an $\ell$ -uniform hypergraph is disperse if the number of edges induced by any set of $\ell +1$ vertices is 0, 1, $\ell$ , or $\ell +1$ . We show that every disperse $\ell$ -uniform hypergraph on $n$ vertices contains a clique or independent set of size $n^{\Omega _{\ell }(1)}$ , answering a question of the first author and Tomon. To this end, we prove several structural properties of disperse hypergraphs.