Infinitely many accumulation points of codegree Turán densities

TL;DR

The paper studies the sets of codegree Turán densities for k-graphs and their accumulation points. It proves that for every $k\ge3$ and $r\ge1$, the point $\frac{r-1}{r}$ is an accumulation point of $\Gamma^{(k)}$, establishing infinitely many accumulation points. The proof uses an inductive construction of $k$-graphs $G_{\ell}^{r}$ built from zycl es and modular families $F_p^{(k)}(n)$ inside a partitioned host to force upper and lower bounds on $\gamma$. This advances Mubayi–Zhao's program by extending zero-accumulation results to a broad class of accumulation points and provides a versatile framework that blends blow-up, supersaturation, and modular obstructions.

Abstract

The codegree Turán density $γ(F)$ of a $k$-graph $F$ is the smallest $γ\in[0,1)$ such that every $k$-graph $H$ with $δ_{k-1}(H)\geq(γ+o(1))\vert V(H)\vert$ 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 $Γ^{(k)}=\{γ(F):F\text{ is a }k\text{-graph}\}$. This makes progress on a problem posed by Mubayi and Zhao.

Figures (1)

• Figure 3.1: Construction of $G_{\ell}^2$ from $G_{\ell}^1$ for $k=3$

Theorems & Definitions (7)

• Theorem 1.1: PS:23
• Theorem 1.2
• Proposition 2.1
• Definition 2.2
• Lemma 2.3: Lemma 2.1 in PS:23
• Definition 2.4
• Lemma 2.5