On the VC-dimension of uniform hypergraphs

  • Authors:

  • Dhruv Mubayi;Yi Zhao

  • Affiliations:

  • Department of Mathematics, Statistics, and Computer Science, University of Illinois, Chicago, USA 60607;Department of Mathematics and Statistics, Georgia State University, Atlanta, USA 30303

  • Venue:
  • Journal of Algebraic Combinatorics: An International Journal
  • Year:
  • 2007

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Abstract

Let $${\cal F}$$ be a k-uniform hypergraph on [n] where k驴1 is a power of some prime p and n驴 n 0(k). Our main result says that if $$|F| ({n\atop k-1}) -\log_p n + k!k^k $$ , then there exists E 0驴 $${\cal F}$$ such that {E驴 E 0: E驴 $${\cal F}$$ } contains all subsets of E 0. This improves a longstanding bound of $$({n\atop k-1})$$ due to Frankl and Pach [7].