# Extending factorizations of complete uniform hypergraphs

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## 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*≥2*m*, 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.

### Mathematics Subject Classification (2000)

05C70 05C65 05C15## Preview

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