An extremal set theoretical characterization of some Steiner systems (Q790818)

For n, k, t integers and \(n>k>t\geq 0\), let m(n,k,t) denote the maximum number of sets in a family of k-subsets of an n-set, no two of which intersect in exactly t elements. \textit{P. Erdős} [Proc. 5th Br. comb. Conf., Aberdeen 1975, 169-192 (1976; Zbl 0335.05002)] first posed the problem of determining m(n,k,t); conjecturing that for \(k>2t+1\) and \(n>n_ 0(k,t)\), \(m(n,k,t)=\left( \begin{matrix} n-t-1\\ k-t-1\end{matrix} \right).\) (A proof of this has been announced by the author and Z. Füredi.) In this paper the author shows that for \(k\leq 2t+1\) and k-t prime, \(m(n,k,t)\leq\left( \begin{matrix} n\\ t\end{matrix} \right)\left( \begin{matrix} 2k-t-1\\ k\end{matrix} \right)/\left( \begin{matrix} 2k-t-1\\ t\end{matrix} \right);\) with equality holding if and only if an (n,2k-t-1,t)-Steiner system exists. A linear algebraic approach is used.