A bound for the lengths of arcs in Steiner systems (Q1808720)

A (partial) Steiner system \(\text{S}(t,k,v)\) is a pair \((\Omega,\beta)\), where \(\Omega\) is a set of \(v\) points and \(\beta\) is a set of \(k\)-subsets (called blocks) of \(\Omega\), such that any \(t\)-subset of \(\Omega\) is contained in exactly (at most) one block. The greatest number of blocks in \(\text{S}(t,k,v)\) is denoted by \(\phi(t,k,v)\). If there exists a Steiner system \(\text{S}(t,k,v)\), then \(\phi(t,k,v)\cdot\left(\begin{smallmatrix} k\\ t\end{smallmatrix}\right)= \left(\begin{smallmatrix} v\\ t\end{smallmatrix}\right)\). The main result is the following theorem: If a Steiner system \(\text{S}(t,k,v)\) with \(2\leq t< k<v\) has an arc \(M\) of length \(m\), then \(\left(\begin{smallmatrix} m\\ t\end{smallmatrix}\right)(k- t)\leq \phi(t-1,t,m)\cdot(v- m)\), with equality if and only if a partial Steiner system \(\text{PS}(t- 1,t,m)\) determined by \((M,\sigma)\), where \(\sigma= \{M\cap B\mid B\in\beta\), \(p\in B\), \(p\not\in M\), \(|M-B|= t\}\), has the maximum number \(\phi(t- 1,t,m)\) of blocks.