Infinitely many accumulation points of codegree Tur\'an densities

TL;DR

This paper proves that for all integers k ≥ 3 and r ≥ 1, the value (r-1)/r is an accumulation point of the set of codegree Turán densities of k-graphs, advancing understanding of their distribution.

Contribution

The paper establishes that (r-1)/r are accumulation points of codegree Turán densities for all k ≥ 3 and r ≥ 1, addressing a problem by Mubayi and Zhao.

Findings

(r-1)/r are accumulation points of b3(F) for all k e2; 3 and r e2; 1.

Progress on Mubayi and Zhao's problem about the distribution of Ture1n densities.

Advances understanding of the structure of codegree Ture1n densities.

Abstract

The codegree Tur\'an density $\gamma(F)$ of a $k$-graph $F$ is the smallest $\gamma\in[0,1)$ such that every $k$-graph $H$ with $\delta_{k-1}(H)\geq(\gamma+o(1))\vert V(H)\vert$ contains a copy of $F$. We prove that for all $k,r\in\mathbb{N}$ with $k\geq3$, $\frac{r-1}{r}$ is an accumulation point of $\Gamma^{(k)}=\{\gamma(F):F\text{ is a }k\text{-graph}\}$. This makes progress on a problem posed by Mubayi and Zhao.