Asymptotic evaluation of the sat-function for \(r\)-stars (Q1972152)

Let \(S^r_m\) be the \(r\)-uniform set system on \(m\) vertices consisting of all \(r\)-tuples containing a given vertex. The author proves the following result: Let \(m> r\geq 2\) and \(S= S^r_m\). Then \[ {m-r\over 2}\begin{pmatrix} n\\ r-1\end{pmatrix}\geq \text{sat}(n, S)\geq m-\text{sat}(n, S)\geq {m-r\over 2}\begin{pmatrix} n\\ r-1\end{pmatrix}- O(n^{r- 4/3}). \]