VC Dimension and a Union Theorem for Set Systems

Abstract

Fix positive integers k and d. We show that, as n → ∞, any set system A ⊂ 2 [n] for which the VC dimension of {△ k i=1 S i | S i ∈ A} is at most d has size at most (2 d mod k + o(1)) n ⌊d/k⌋ . Here △ denotes the symmetric difference operator. This is a k-fold generalisation of a result of Dvir and Moran, and it settles one of their questions.A key insight is that, by a compression method, the problem is equivalent to an extremal set theoretic problem on k-wise intersection or union that was originally due to Erdős and Frankl.We also give an example of a family A ⊂ 2 [n] such that the VC dimension of A ∩ A and of A ∪ A are both at most d, while |A| = Ω(n d ). This provides a negative answer to another question of Dvir and Moran.It is easy to see that this bound is sharp by taking, for example, A = [n] ≤d . This bound has fundamental importance and wide applicability, e.g. in machine learning, model theory, graph theory, and computational geometry.