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arXiv · 2501.15457

A note on the minimum size of Turán systems

Xizhi Liu·Oleg Pikhurko
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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 $O(1)<s-r = O(r/\ln r)$.

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Xizhi Liu, Oleg Pikhurko. 2025-02-02. A note on the minimum size of Turán systems. https://arxiv.org/abs/2501.15457

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