Generating all subsets of a finite set with disjoint unions

Authors:
David Ellis;Benny Sudakov
Affiliations:
St Johns College, Cambridge, CB2 1TP, United Kingdom;Department of Mathematics, UCLA, Los Angeles, CA 90095, United States
Venue:
Journal of Combinatorial Theory Series A
Year:
2011

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Abstract

If X is an n-element set, we call a family G@?PX a k-generator for X if every x@?X can be expressed as a union of at most k disjoint sets in G. Frein, Leveque and Sebo conjectured that for n2k, 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=3.