Generating all subsets of a finite set with disjoint unions
Combinatorics
2011-06-06 v2
Abstract
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.
Cite
@article{arxiv.1010.4877,
title = {Generating all subsets of a finite set with disjoint unions},
author = {David Ellis and Benny Sudakov},
journal= {arXiv preprint arXiv:1010.4877},
year = {2011}
}
Comments
Final version, with some additional explanations added in the proofs