Separating hypergraph Turán densities

TL;DR

This work addresses the long-standing problem of distinguishing Turán densities for clique hypergraphs in $k$-uniform systems by introducing a general, flag-algebra-free separation criterion. The authors prove $\pi(K_{\ell}^{(k)})<\pi(K_{\ell+1}^{(k)})$ for all $\ell>k\ge 3$ and $\pi(K_{\ell}^{(k)-})<\pi(K_{\ell+1}^{(k)-})$, and deduce $\pi(K_{k+1}^{(k)})<\pi(K_{k+2}^{(k)-})$ for all $k\ge 3$, with the $k=3$ case aligning with Markström's result. They provide a quick negative answer to Erdős's problem via a simple bound, and, crucially, present a new computer-free lower bound for the $k=3$ case using a novel construction that yields $\pi(K_4^{(3)})<\pi(K_5^{(3)-})$ by achieving $0.602673\ldots$ density. Overall, the paper offers a versatile framework for separating hypergraph Turán densities and poses natural open questions about broader families such as daisies $H_t^{(k)}$ and beyond.

Abstract

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

Theorems & Definitions (14)

• Theorem 1.1
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Corollary 1.6
• Proposition 2.1
• proof
• Lemma 2.2
• proof
• proof : Proof of Theorem \ref{['THM:general-result']}