Exponents of uniform \(L\)-systems (Q1919666)

An \((n, k, L)\)-system, \(L\subset [0,k-1]\), is a family \({\mathcal F}\subset(\begin{smallmatrix} [n]\\ k\end{smallmatrix})\) such that \(|F\cap F'|\in L\) for all distinct \(F,F'\in {\mathcal F}\). If there exist constants \(c\), \(c'\), \(\alpha\), depending only on \(k\) and \(L\), satisfying \[ cn^\alpha<\max\{|{\mathcal F}|:{\mathcal F}\text{ is an }(n, k, L)\text{-system}\}<c'n^\alpha, \] then \(\alpha=\alpha(k,L)\) is called the exponent of the \((n, k, L)\)-system. The authors determine all such exponents for \(k\leq12\), with two exceptions related to the Steiner systems \(S(4, 5, 11)\) and \(S(5, 6, 12)\). Previously known results and constructions are reviewed. New constructions are provided, examples are given and exponents are summarized in tables.