Baranyai's theorem states that if , then the complete -uniform hypergraph on vertices decomposes into 1-factors, where
a 1-factor is a set of pairwise disjoint -sets. Brouwer and Schrijver (1979) give a beautiful proof
using the max-flow min-cut theorem of
network flows.
Baranyai, Z. "On the Factorization of the Complete Uniform Hypergraph. Infinite and Finite Sets." In Infinite and Finite Sets,
Vol. 1. Proceedings of a Colloquium Held at Keszthely, June 25-July 1, 1973.
Dedicated to Paul Erdős on His 60th Birthday (Ed. A. Hajnal, R. Rado,
and V. T. Sós). Amsterdam, Netherlands: North-Holland, pp. 91-108,
1975.Brouwer, A. E. and Schrijver, A. "Uniform Hypergraphs."
In Packing
and Covering in Combinatorics. Mathematical Centre Tracts, No. 106,
pp. 39-73, 1979.Tamm, U. "Applications of Baranyai's Theorem
in Information Theory." In Proceedings of 6th Benelux-Japan Workshop on Coding
and Information Theory, Essen, 1996 (Ed. A. J. Han Vinck and A. van
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Lint, J. H. and Wilson, R. M. A
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1993.West, D. "Re: disjoint cliques, resolutions of designs?"
GRAPHNET@listserv.nodak.edu posting. Feb. 25, 2004. https://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0402&L=graphnet&F=&S=&P=4041.
, then the complete 
-uniform hypergraph on 
vertices decomposes into 1-factors, where
a 1-factor is a set of 
pairwise disjoint 
-sets. Brouwer and Schrijver (1979) give a beautiful proof
using the max-flow min-cut theorem of
network flows.
