A note on the minimum size of Turán systems

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 $s-r=\mathrm{\Omega }(r\mathrm{/}\ln r)$ (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 $O(1)<s-r=O(r\mathrm{/}\ln r)$.