Small transversals in hypergraphs

Abstract

For each positive integerk, we consider the setA k of all ordered pairs [a, b] such that in everyk-graph withn vertices andm edges some set of at mostam+bn vertices meets all the edges. We show that eachA k withk≥2 has infinitely many extreme points and conjecture that, for every positive ε, it has only finitely many extreme points [a, b] witha≥ε. With the extreme points ordered by the first coordinate, we identify the last two extreme points of everyA k , identify the last three extreme points ofA 3, and describeA 2 completely. A by-product of our arguments is a new algorithmic proof of Turán's theorem.

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Chvátal, V., McDiarmid, C. Small transversals in hypergraphs. Combinatorica 12, 19–26 (1992). https://doi.org/10.1007/BF01191201

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AMS subject classification (1991)

  • 05 C 65
  • 05 B 40
  • 05 D 05