Rational values and non-principality for $\ell$-degree Turán densities of hypergraphs

Abstract

Let $k>\ell\ge 1$ be integers. For a family $\mathcal{F}$ of $k$-uniform hypergraphs ($k$-graphs), the $\ell$-degree Turán density $γ^{(k)}_\ell(\mathcal{F})$ of $\mathcal{F}$ is defined as the asymptotic maximum of the normalized minimum $\ell$-degree over all $\mathcal{F}$-free $k$-graphs. In this paper, we prove that for all $k>\ell>k/2$, every rational number $α\in[0,1)$ can be realized as the $\ell$-degree Turán density $γ^{(k)}_\ell(\mathcal{F})$ for some finite family $\mathcal{F}$ of $k$-graphs. Furthermore, for any $k>\ell\ge 1$, we construct an explicit infinite sequence of values realized by single forbidden $k$-graphs: for each integer $q\ge2$, there exists a $k$-graph $F$ such that $γ^{(k)}_\ell(F)=1-q^{\ell-k}$. We also establish a strengthened non-principality property for $\ell$-degree Turán densities: for all $k>\ell>1$, there exist two $k$-graphs $F_1$ and $F_2$ satisfying $0< γ^{(k)}_{\ell}(\{F_1,F_2\})<\min\{ γ^{(k)}_\ell(F_1),γ^{(k)}_\ell(F_2)\}$.