On asymptotic local Turán problems
Peter Frankl, Jiaxi Nie
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 Rödl 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 $γ\ge1$ that $\lim_{q\to\infty}t_r((q,\lceil{q/γ}\rceil))=1/\lfloorγ\rfloor^{r-1}$. 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.