Constrained versions of Sauer's lemma

Joel Ratsaby

Research output: Contribution to journalArticlepeer-review


Let [n] = {1, ..., n}. For a function h : [n] → {0, 1}, x ∈ [n] and y ∈ {0, 1} define by the widthωh (x, y) of h at x the largest nonnegative integer a such that h (z) = y on x - a ≤ z ≤ x + a. We consider finite VC-dimension classes of functions h constrained to have a width ωh (xi, yi) which is larger than N for all points in a sample ζ = {(xi, yi)}1 or a width no larger than N over the whole domain [n]. Extending Sauer's lemma, a tight upper bound with closed-form estimates is obtained on the cardinality of several such classes.

Original languageEnglish
Pages (from-to)2753-2767
Number of pages15
JournalDiscrete Applied Mathematics
Issue number14
StatePublished - 28 Jul 2008
Externally publishedYes


  • Binary functions
  • Integer partitions
  • Sauer's lemma

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Applied Mathematics


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