Covering triples by quadruples: an asymptotic solution (Q1072560)

\textit{J. Schoenheim} [Pac. J. Math. 14, 1405--1411 (1964; Zbl 0128.24501)] showed that to cover all triples of an \(n\)-set by quadruples, at least \(\lceil \frac{1}{4}n\lceil \frac{1}{3}(n-1)\lceil \frac{1}{2}(n-2)\rceil \rceil \rceil\) quadruples are needed with equality e.g. when a Steiner quadruple system on n points exists. Using a construction of Mills for \(n=499\) and some recursive techniques it is shown that this number of quadruples suffices for all \(n\geq 52423\).