Intersecting sperner families and their convex hulls

Abstract

Let ℱ be a family of subsets of a finite set ofn elements. The vector (f 0, ...,f n ) is called the profile of ℱ wheref i denotes the number ofi-element subsets in ℱ. Take the set of profiles of all families ℱ satisfyingF 1F 2 andF 1F 2≠0 for allF 1,F 2teℱ. It is proved that the extreme points of this set inR n+1 have at most two non-zero components.

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Dedicated to Paul Erdős on his seventieth birthday

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Erdős, P.L., Frankl, P. & Katona, G.O.H. Intersecting sperner families and their convex hulls. Combinatorica 4, 21–34 (1984). https://doi.org/10.1007/BF02579153

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AMS subject classification (1980)

  • 05 C 35
  • 05 C 65
  • 52 A 20