Some inequalities for Steiner systems (Q1266399)

A partial Steiner system \(\text{PS}(t,k,v)\) is a pair \((\Omega, B)\), \(\Omega\) being a set of \(v\) elements and \(B\) being a set of \(k\)-subsets (called blocks) of \(\Omega\) such that any \(t\)-subset of \(\Omega\) is contained in at most one block. \(\text{PS}(t,k,v)\) is called a Steiner system \(\text{S}(t,k,v)\) if any \(t\)-subset is contained in exactly one block. The following results are presented: For Steiner systems \(\text{S}(t,k,v)\) with \(2\leq t< k< v\), the following results are true: \[ {k\choose t-1}(k- t)\leq \Biggl[{k\over t-1} \Biggl[{k-1\over t-2} \Biggl[\dots \Biggl[{k- t+4\over 3} \Biggl[{k- t+3\over 2}\Biggr]\Biggr]\dots\Biggr]\Biggr]\Biggr](v- k-1),\tag{1} \] \[ {k\choose k- t+1}(k- t)\leq \Biggl[{k\over k- t+1} \Biggl[{k-1\over k-t} \Biggr[\dots \Biggr[{t+2\over 3} \Biggl[{t+ 1\over 2}\Biggr]\Biggr]\dots\Biggr]\Biggr]\Biggr](v- k-1).\tag{2} \] Here \([m]\) stands for the maximal integer not exceeding \(m\). Results (1) and (2) improve Fisher's inequality \((v- t+1)\geq (k- t+ 1)(k- t+ 2)\) and the inequality \(v\geq (t+ 1)(k- t+ 1)\).