Unavoidable Traces Of Set Systems

Sauer, Shelah, Vapnik and Chervonenkis proved that if a set system on n vertices contains many sets, then the set system has full trace on a large set. Although the restriction on the size of the groundset cannot be lifted, Frankl and Pach found a trace structure that is guaranteed to occur in uniform set systems even if we do not bound the size of the groundset. In this note we shall give three sequences of structures such that every set system consisting of sufficiently many sets contains at least one of these structures with many sets.

This is a preview of subscription content, access via your institution.

Author information



Corresponding authors

Correspondence to József Balogh or Béla Bollobás.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Balogh, J., Bollobás, B. Unavoidable Traces Of Set Systems. Combinatorica 25, 633–643 (2005). https://doi.org/10.1007/s00493-005-0039-x

Download citation

Mathematics Subject Classification (2000):

  • 05C35
  • 05C65
  • 05D05