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 ${\gamma }_{\ell }^{(k)}(\mathcal{F})$ 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 $\alpha \in [0,1)$, there exists an infinite family $\mathcal{F}$ of $k$-graphs such that ${\gamma }_{\ell }^{(k)}(\mathcal{F})=\alpha$, while every finite subfamily ${\mathcal{F}}^{\prime }\subset \mathcal{F}$ satisfies ${\gamma }_{\ell }^{(k)}({\mathcal{F}}^{\prime })>\alpha$. This confirms the conjecture of Mubayi and Zhao on codegree Turán densities for every $k>3$. As an immediate consequence, the set ${\Gamma }_{\infty }^{k,\ell }$ 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.

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