Separating hypergraph Turán densities

Abstract

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

11 pages, updated introduction