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Let $M$ be a set with $2\;|\;M$. What is the number of mutually disjoint partitions, each consisting of subsets with a cardinality of $2$.

$|M|-1$ should be a trivial upper bound.

Note: I asked the same question yesterday, although I accidently deleted it.

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you can undelete your question I believe –  Ilya Oct 15 '11 at 19:37
    
@Gortaur: I could not figure it out. –  Ioswa Oct 15 '11 at 19:39
    
I cannot check it on mine questions - I guess they're too old. I thought I could do it - so sorry for the confusion –  Ilya Oct 15 '11 at 19:45

1 Answer 1

up vote 3 down vote accepted

By a very special case (known for a century or two) of Baranyai's theorem, the edges of a complete graph on $M$ vertices may be decomposed into $M-1$ perfect matchings (when $M$ is even). These matchings give you the desired partitions. There's a picture in the wikipedia link that indicates the general construction of the edge decomposition.

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