Abstract
We show that any family of subsets $A\subseteq 2^{[n]}$ satisfies $\lvert A\rvert \leq O\bigl(n^{\lceil{d}/{2}\rceil}\bigr)$, where $d$ is the VC dimension of $\{S\triangle T \,\vert\, S,T\in A\}$, and $\triangle$ is the symmetric difference operator. We also observe that replacing $\triangle$ by either $\cup$ or $\cap$ fails to satisfy an analogous statement. Our proof is based on the polynomial method; specifically, on an argument due to [Croot, Lev, Pach '17].
Highlights
IntroductionThe VC dimension of A, denoted by VC-dim(A), is the size of the largest Y ⊆ [n] such that {S∩Y | S ∈ A} = 2Y
Let A ⊂ 2[n] be a family of subsets of an n element set ([n] w.l.o.g)
In this work we explore the converse direction: Does an upper bound on the VC-dimension VC-dim(A A) imply an upper bound on |A|? It is not hard to see that VC-dim(A) VC-dim(A A) for ∈ {∪, ∩, }, and by Theorem 1: VC-dim(A A) < d =⇒ |A| O(nd)
Summary
The VC dimension of A, denoted by VC-dim(A), is the size of the largest Y ⊆ [n] such that {S∩Y | S ∈ A} = 2Y. One of the most useful facts about the VC dimension is given by the Sauer-Shelah-Perles Lemma. A simple-yet-useful corollary of this lemma is that if VC-dim(A) d, and is any binary set-operation Suppose A ⊂ 2[n] satisfies VC-dim(A A) d. The above examples rule out the analog of Theorem 2 for exactly one of ∪, ∩. This suggests the following open question: Question 3. D and Another natural question is whether this phenomenon extends to several applications of the symmetric difference operator, for example: Question 4.
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