Hypergraphs accumulate

TL;DR

This paper demonstrates that the set of Turán densities for k-uniform hypergraphs has accumulation points within [0,1), highlighting complex density behaviors, especially noting 1/2 as an accumulation point for 3-uniform hypergraphs.

Contribution

It establishes the existence of accumulation points in the set of Turán densities for all k ≥ 3, revealing new structural properties of hypergraph densities.

Findings

Turán densities have accumulation points in [0,1)

1/2 is an accumulation point for 3-uniform hypergraphs

The set of densities exhibits complex limiting behaviors

Abstract

We show that for every integer $k\geq3$, the set of Tur\'an densities of $k$-uniform hypergraphs has an accumulation point in $[0,1)$. In particular, $1/2$ is an accumulation point for the set of Tur\'an densities of $3$-uniform hypergraphs.