Disperse Hypergraphs

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.