$\ell$-degree Turán density

Abstract

Let $H_n$ be a $k$-graph on $n$ vertices. For $0\le \ell <k$ and an $\ell$-subset $T$ of $V(H_n)$, define the degree $°(T)$ of $T$ to be the number of $(k-\ell )$-subsets~$S$ such that $S\cup T$ is an edge in~$H_n$. Let the minimum $\ell$-degree of $H_n$ be ${\delta }_{\ell }(H_n)=\min \{°(T):T\subseteq V(H_n)$ and $|T|=\ell \}$. Given a family $\mathcal{F}$ of $k$-graphs, the $\ell$-degree Turán number ${\text{ex}}_{\ell }(n,\mathcal{F})$ is the largest ${\delta }_{\ell }(H_n)$ over all $\mathcal{F}$-free $k$-graphs $H_n$ on $n$ vertices. Hence, ${\text{ex}}_0(n,\mathcal{F})$ is the Turán number. We define $\ell$-degree Turán density to be $${\pi }_{\ell }^k(\mathcal{F})=\underset{n\to \infty }{\mathrm{limsup}}\frac{{\text{ex}}_{\ell }(n,\mathcal{F})}{\left(\genfrac{}{}{0ex}{}{n-\ell }{k}\right)}.$$ In this paper, we show that for $k>\ell >1$, the set of ${\pi }_{\ell }^k(\mathcal{F})$ is dense in the interval $[0,1)$. Hence, there is no "jump" for $\ell$-degree Turán density when $k>\ell >1$. We also give a lower bound on ${\pi }_{\ell }^k(\mathcal{F})$ in terms of an ordinary Turán density.

Updated. Now published in SIAM J. Discrete Math. 28 (2014), 1214-1225