Note on generating all subsets of a finite set with disjoint unions

2008/11/18 by Ellis, David
#05D05 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.0811.3022

Abstract

We call a family G of subsets of [n] a k-generator of (ℙ[n]) if every (x ⊂ [n]) can be expressed as a union of at most k disjoint sets in (G). Frein, Leveque and Sebo conjectured that for any (n ≥ k), such a family must be at least as large as the k-generator obtained by taking a partition of [n] into classes of sizes as equal as possible, and taking the union of the power-sets of the classes. We generalize a theorem of Alon and Frankl \citealon in order to show that for fixed k, any k-generator of (ℙ[n]) must have size at least (k2^n/k(1-o(1))), thereby verifying the conjecture asymptotically for multiples of k.

Related

• Generating all subsets of a finite set with disjoint unions
• Minimizing the Number of Unions
• Optimizing diversity
• Union-Free Families of Subsets
• Minimum-sized generating sets of the direct powers of free distributive lattices