$\mathrm{\ell }$-degree Tur\'an density

Let $H_n$ be a $k$-graph on $n$ vertices. For $0\le \mathrm{\ell }<k$ and an $\mathrm{\ell }$-subset $T$ of $V(H_n)$, define the degree $\mathrm{deg}(T)$ of $T$ to be the number of $(k-\mathrm{\ell })$-subsets~$S$ such that $S\cup T$ is an edge in~$H_n$. Let the minimum $\mathrm{\ell }$-degree of $H_n$ be $\delta_{\ell}(H_n) = \min { \deg(T) : T \subseteq V(H_n)$and$|T|=\ell}$\mathrm{.}Givenafamily$$\mathcal{F}$$of$k$-graphs, the $\mathrm{\ell }$-degree Tur'an number ${\text{ex}}_{\mathrm{\ell }}(n,\mathcal{F})$ is the largest ${\delta }_{\mathrm{\ell }}(H_n)$ over all $\mathcal{F}$-free $k$-graphs $H_n$ on $n$ vertices. Hence, ${\text{ex}}_0(n,\mathcal{F})$ is the Tur'an number. We define $\mathrm{\ell }$-degree Tur'an density to be $${\pi }_{\mathrm{\ell }}^k(\mathcal{F})=\underset{n\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}sup}}\frac{{\text{ex}}_{\mathrm{\ell }}(n,\mathcal{F})}{\left(\genfrac{}{}{0px}{}{n-\mathrm{\ell }}{k}\right)}\mathrm{.}$$ In this paper, we show that for $k>\mathrm{\ell }>1$, the set of ${\pi }_{\mathrm{\ell }}^k(\mathcal{F})$ is dense in the interval $[0,1)$. Hence, there is no “jump” for $\mathrm{\ell }$-degree Tur'an density when $k>\mathrm{\ell }>1$. We also give a lower bound on ${\pi }_{\mathrm{\ell }}^k(\mathcal{F})$ in terms of an ordinary Tur'an density.