Disperse Hypergraphs

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