On the Limit of the Positive $\ell$-Degree Turán Problem

Abstract

The minimum positive $\ell$-degree $\delta^+_{\ell}(G)$ of a non-empty $k$-graph $G$ is the maximum $m$ such that every $\ell$-subset of $V(G)$ is contained in either none or at least $m$ edges of~$G$; let $\delta^+_{\ell}(G):=0$ if $G$ has no edges. For a family $\mathcal F$ of $k$-graphs, let $\mathrm{co^{+}ex}_\ell(n,\mathcal F)$ be the maximum of $\delta^+_{\ell}(G)$ over all $\mathcal F$-free $k$-graphs $G$ on $n$ vertices. We prove that the ratio $\mathrm{co^{+}ex}_\ell(n,\mathcal F)/{n-\ell\choose k-\ell}$ tends to limit as $n\to\infty$, answering a question of Halfpap, Lemons and Palmer. Also, we show that the limit can be obtained as the value of a natural optimisation problem for $k$-hypergraphons; in fact, we give an alternative description of the set of possible accumulation points of almost extremal $k$-graphs.