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Extending factorizations of complete uniform hypergraphs. (English) Zbl 1438.05197
Summary: We consider when a given $$r$$-factorization of the complete uniform hypergraph on $$m$$ vertices $$K^h_m$$ can be extended to an $$s$$-factorization of $$K^h_n$$. The case of $$r = s = 1$$ was first posed by P. J. Cameron [Parallelisms of complete designs. Cambridge: Cambridge University Press; London: London Mathematical Society (1976; Zbl 0333.05007)] in terms of parallelisms, and solved by R. Häggkvist and T. Hellgren [in: Combinatorics, Paul Erdős is eighty. Vol. 1. Budapest: János Bolyai Mathematical Society. 215–238 (1993; Zbl 0795.05053)]. We extend these results, which themselves can be seen as extensions of the theorem of Zs. Baranyai [in: Infinite and finite sets. To Paul Erdős on his 60th birthday. Vols. I, II, III. Amsterdam etc.: North-Holland Publishing Company. 91–108 (1975; Zbl 0306.05137)]. For $$r=s$$, we show that the “obvious” necessary conditions, together with the condition that $$\operatorname{gcd}(m,n,h)=\operatorname{gcd}(n,h)$$ 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 $$\operatorname{gcd}(m,n,h)=\operatorname{gcd}(n,h)$$, $$n\geq 2m$$, and $$1\leq \frac{s}{r}\leq \frac{m}{k}[1-\binom{m-k}{h}/\binom{m}{h}]$$ are sufficient, where $$k=\operatorname{gcd}(m,n,h)$$. We conclude with a discussion of further necessary conditions and some open problems.
##### MSC:
 05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.) 05C65 Hypergraphs 05C15 Coloring of graphs and hypergraphs
##### Keywords:
factorization; uniform hypergraph
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