The $\ell$-degree Turán densities of hypergraphs: infinite families and their finite subfamilies

Abstract

Let $k>\ell\ge 1$ be integers. For a family $\mathcal{F}$ of $k$-uniform hypergraphs ($k$-graphs), its $\ell$-degree Turán density $γ^{(k)}_\ell(\mathcal{F})$ is the asymptotic maximum of the normalized minimum $\ell$-degree over all $\mathcal{F}$-free $k$-graphs. In this paper, we prove that for all integers $k>\ell>2$ and every $α\in[0,1)$, there exists an infinite family $\mathcal{F}$ of $k$-graphs such that $γ^{(k)}_\ell(\mathcal{F})=α$, while every finite subfamily $\mathcal{F}'\subset \mathcal{F}$ satisfies $γ^{(k)}_\ell(\mathcal{F}')>α$. This confirms the conjecture of Mubayi and Zhao on codegree Turán densities for every $k > 3$. As an immediate consequence, the set $Γ^{k,\ell}_{\infty}$ of $\ell$-degree Turán densities of nonempty $k$-graph families is $[0,1)$ whenever $k>\ell>2$, which answers a question of Lo and Markström from 2014.