Packing nearly-disjoint sets

Abstract

De Bruijn and Erdős proved that ifA 1, ...,A k are distinct subsets of a set of cardinalityn, and |A i A j |≦1 for 1≦i<jk, andk>n, then some two ofA 1, ...,A k have empty intersection. We prove a strengthening, that at leastk /n ofA 1, ...,A k are pairwise disjoint. This is motivated by a well-known conjecture of Erdőds, Faber and Lovász of which it is a corollary.

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Partially supported by N. S. F. grant No. MCS—8103440

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Seymour, P.D. Packing nearly-disjoint sets. Combinatorica 2, 91–97 (1982). https://doi.org/10.1007/BF02579285

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AMS subject classification (1980)

  • 05 C 65
  • 05 C 15