Extending factorizations of complete uniform hypergraphs

Abstract

We consider when a given r-factorization of the complete uniform hypergraph on m vertices K _m ^h can be extended to an s-factorization of K _n ^h . The case of r = s = 1 was first posed by Cameron in terms of parallelisms, and solved by Häggkvist and Hellgren. We extend these results, which themselves can be seen as extensions of Baranyai's Theorem. For r=s, we show that the “obvious” necessary conditions, together with the condition that gcd(m,n,h)=gcd(n,h) are sufficient. In particular this gives necessary and sufficient conditions for the case where r=s and h is prime. For r<s we show that the obvious necessary conditions, augmented by gcd(m,n,h)=gcd(n,h), n≥2m, and \(1 \leqslant \frac{s}{r} \leqslant \frac{m}{k}\left[ {1 - \left( {\begin{array}{*{20}{c}} {m - k} \\ h \end{array}} \right)/\left( {\begin{array}{*{20}{c}} m \\ h \end{array}} \right)} \right]\) are sufficient, where k=gcd(m,n,h). We conclude with a discussion of further necessary conditions and some open problems.