Hypergraphs accumulate infinitely often

TL;DR

This paper proves that the set of Turán densities for k-uniform hypergraphs has infinitely many accumulation points within the interval [0,1) for all k ≥ 3, extending previous results.

Contribution

It establishes that the set of Turán densities for k-uniform hypergraphs has infinitely many accumulation points, generalizing earlier findings.

Findings

The set of Turán densities has infinitely many accumulation points.

This holds for all k ≥ 3.

It extends previous results on the structure of Turán densities.

Abstract

We show that the set $\Pi^{(k)}$ of Tur\'an densities of $k$-uniform hypergraphs has infinitely many accumulation points in $[0,1)$ for every $k \geq 3$. This extends an earlier result of ours showing that $\Pi^{(k)}$ has at least one such accumulation point.