Generating all sets with bounded unions

  • Authors:

  • Yannick Frein;Benjamin LÉvÊque;AndrÁs SebŐ

  • Affiliations:

  • Laboratoire g-scop, inpg, ujf, cnrs, 46, avenue felix viallet, 38031 grenoble cedex, france (e-mail: yannick.frein.benjamin.leveque.andras.sebo@g-scop.inpg.fr);Laboratoire g-scop, inpg, ujf, cnrs, 46, avenue felix viallet, 38031 grenoble cedex, france (e-mail: yannick.frein.benjamin.leveque.andras.sebo@g-scop.inpg.fr);Laboratoire g-scop, inpg, ujf, cnrs, 46, avenue felix viallet, 38031 grenoble cedex, france (e-mail: yannick.frein.benjamin.leveque.andras.sebo@g-scop.inpg.fr)

  • Venue:
  • Combinatorics, Probability and Computing
  • Year:
  • 2008

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Abstract

We consider the problem of minimizing the size of a family of sets such that every subset of {1,…, n} can be written as a disjoint union of at most k members of , where k and n are given numbers. This problem originates in a real-world application aiming at the diversity of industrial production. At the same time, the question of finding the minimum of | | so that every subset of {1,…, n} is the union of two sets in was asked by ErdQs and studied recently by Füredi and Katona without requiring the disjointness of the sets. A simple construction providing a feasible solution is conjectured to be optimal for this problem for all values of n and k and regardless of the disjointness requirement; we prove this conjecture in special cases including all (n, k) for which n≤3k holds, and some individual values of n and k.