Generating all subsets of a finite set with disjoint unions

Details Generating all subsets of a finite set with disjoint unions Generating all subsets of a finite set with disjoint unions

2010-10-23

arxiv.org/abs/1010.4877v2

Mathematics

Combinatorics

Final version, with some additional explanations added in the proofs

Scientific paper

If X is an n-element set, we call a family G of subsets of X a k-generator for X if every subset of X can be expressed as a union of at most k disjoint sets in G. Frein, Leveque and Sebo conjectured that for n > 2k, the smallest k-generators for X are obtained by taking a partition of X into classes of sizes as equal as possible, and taking the union of the power-sets of the classes. We prove this conjecture for all sufficiently large n when k = 2, and for n a sufficiently large multiple of k when k > 2.

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