Separating hypergraph Tur\'an densities

TL;DR

This paper proves that the Turán densities of complete hypergraphs increase with the number of vertices for fixed uniformity, introduces a criterion to distinguish hypergraph densities, and resolves a longstanding conjecture for specific cases.

Contribution

It establishes the strict inequality of Turán densities for hypergraphs with increasing size and provides a general criterion for comparing hypergraph densities, advancing understanding of hypergraph extremal problems.

Findings

Proved $ ext{pi}(K_{ ext{l}}^{( ext{k})})< ext{pi}(K_{ ext{l+1}}^{( ext{k})})$ for all $ ext{l}>k ext{≥}3$.

Introduced a criterion to distinguish Turán densities of hypergraphs.

Confirmed a special case previously conjectured by Erdős.

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.