A note on the minimum size of Turán systems
Xizhi Liu, Oleg Pikhurko
TL;DR
This paper studies Turán $(n,s,r)$-systems, where every $s$-subset of an $n$-vertex $r$-graph contains an edge, and focuses on upper bounds for the minimum size $T(n,s,r)$ through the asymptotic density $t(s,r)$ and the derived parameter $\mu(s,r)=t(s,r)\binom{s}{r}$. Building on prior work that handles the extremes $s-r=Ω(r/\ln r)$ and $s-r=O(1)$, the authors address the intermediate regime $1< s-r=O(r/\ln r)$ by proving near-optimal bounds: for any $ε>0$ there exists $r_0$ such that if $R=s-r\ge r_0$, then $\mu(r+R,r) \le (1+ε)R\ln\binom{r+R}{R}$, and a complementary bound $\mu(r+R,r) \le e^{18R^2/r}(1+ε)R\ln R$ when $R\le\sqrt{18r\ln r}$. The method combines a Frankl–Rödl–style probabilistic construction with the Lovász Local Lemma to produce a base Turán system on $[N]$ and a blowup to a larger $n$, yielding explicit control over $|\mathcal{B}|$ and ultimately the $\mu$-bounds. These results extend and unify previous bounds, bridging Sidorenko’s regime and Pik24’s recursion to cover a broad intermediate range of $R$ and improving our understanding of Turán-type densities in high uniformity hypergraphs. $t(s,r)$, $T(n,s,r)$, $\mu(s,r)$, and the intermediate regime $R=s-r$ are central to the analysis, with the Local Lemma and recursive inequalities providing the key technical backbone. $
Abstract
For positive integers $n \ge s > r$, a \emph{Turán $(n,s,r)$-system} is an $n$-vertex $r$-graph in which every set of $s$ vertices contains at least one edge. Let $T(n,s,r)$ denote the the minimum size of a Turán $(n,s,r)$-system. Upper bounds on $T(n,s,r)$ were established by Sidorenko~\cite{Sid97} for the case $s-r = Ω(r/\ln r)$ (based on a construction of Frankl--Rödl~\cite{FR85}) and by a number of authors in the case $s-r = O(1)$. In this note, we establish upper bounds in the remaining range $O(1)<s-r = O(r/\ln r)$.