On asymptotic local Tur\'an problems

Abstract

An $r$-uniform hypergraph has $(q,p)$-property if any set of $q$ vertices spans a complete sub-hypergraph on $p$ vertices. Let $t_r(n,q,p)$ be the minimum edge density of an $n$-vertex $r$-uniform hypergraph with {\em $(q,p)$-property} and let $t_r(q,p)=\lim_{n\to\infty}t_r(n,q,p)$. A disjoint union of $k$ complete hypergraphs has $(q,\lceil q/k\rceil)$-property, which gives $t_r((q,\lceil{q/k}\rceil))\le 1/k^{r-1}$. The first author, Huang and R\"odl showed that these constructions are the best asymptotically, that is, $\lim_{q\to\infty}t_r((q,\lceil{q/k}\rceil))=1/k^{r-1}$. They asked whether it is true for all real number $\gamma\ge1$ that $\lim_{q\to\infty}t_r((q,\lceil{q/\gamma}\rceil))=1/\lfloor{\gamma}\rfloor^{r-1}$. In this paper, we give positive answers to this question for a small range of real numbers, and, on the other hand, provide new constructions that give negative answers for many other ranges.