A note on the minimum size of Tur\'{a}n systems

TL;DR

This paper investigates the minimum size of Turán systems for certain parameters, providing new upper bounds in a previously unaddressed range of the difference between s and r.

Contribution

It establishes upper bounds on T(n,s,r) for the case where s-r is between constant and order r/ln r, filling a gap in existing research.

Findings

New upper bounds for T(n,s,r) in the range O(1)<s-r=O(r/ln r)

Extension of previous bounds to a previously unaddressed parameter range

Improved understanding of Turán systems' minimal sizes

Abstract

For positive integers $n \ge s > r$, a \emph{Tur\'{a}n $(n,s,r)$-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 Tur\'{a}n $(n,s,r)$-system. Upper bounds on $T(n,s,r)$ were established by Sidorenko~\cite{Sid97} for the case $s-r = \Omega(r/\ln r)$ (based on a construction of Frankl--R\"{o}dl~\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 $O(1)<s-r = O(r/\ln r)$.