Separating hypergraph Tur\'an densities

Abstract

Determining the Tur\'an densities of hypergraphs is a notoriously difficult problem at the core of combinatorics. Although Tur\'an posed this problem in 1941, $\pi(K_{\ell}^{(k)})$ remains unknown for all $\ell>k\geq 3$. Prior to this work, it was not even known whether $\pi(K_{\ell}^{(k)})<\pi(K_{\ell+1}^{(k)})$ holds for general $\ell$ and $k$, and the best-known bounds on $\pi(K_{\ell}^{(k)})$ are far from implying anything close to this. We prove that $\pi(K_{\ell}^{(k)})<\pi(K_{\ell+1}^{(k)})$, for all $\ell>k\geq 3$, and provide a general criterion to distinguish the Tur\'an densities of two hypergraphs. As a corollary, we obtain that $\pi(K_{k+1}^{(k)})<\pi(K_{k+2}^{(k)-})$, for all $k\geq 3$. For $k=3$, this was previously proved by Markstr\"om, answering a question by Erd\H{o}s.