A Sauer-Shelah-Perles Lemma for Sumsets

Abstract

We extend the Sauer-Shelah-Perles lemma to an abstract setting that is formalized using the language of lattices. Our extension applies to all finite lattices with nonvanishing Möbius function, a rich class of lattices which includes all geometric lattices (or matroids) as a special case.For example, our extension implies the following result in Algebraic Combinatorics: let F be a family of subspaces of F n q . We say that F shatters a subspace U if for every subspace U ′ ≤ U there is F ∈ F such that F ∩U = U ′ . Then, if |F| > n 0 q +· · ·+ n d q then F shatters some (d+1)-dimensional subspace (where n k q denotes the number of k-dimensional subspaces in

VC Dimension and a Union Theorem for Set Systems

“…For every A ⊂ 2 [n] with VC-dim(A△A) ≤ d, we have |A| ≤ 2 r n−r ≤⌊d/2⌋ . This is a best possible form of the result of Dvir and Moran [2].…”

“…We also write ⋆A k = {S 1 ⋆ · · · ⋆ S k | S i ∈ A, ∀i ∈ [k]}. Motivated by an application in PAC learnability, Dvir and Moran [2] recently investigated how large A can be assuming A ⋆ A = {S ⋆ T | S, T ∈ A} has bounded VC dimension. Using the polynomial method, they proved that |A| ≤ 2 n ≤⌊d/2⌋ provided VC-dim(A△A) ≤ d. They also asked whether an analogous result might hold assuming VC-dim(△A k ) ≤ d, particularly for k = 3 [2,Qu.…”

A lower bound for the k‐multicolored sum‐free problem in Zmn

“…Their proof uses a new polynomial method developed by Croot, Lev, and Pach for the analogous problem in . The preprint of Ellenberg and Gijswijt appeared just a few weeks after the one of Croot, Lev, and Pach, and subsequently a lot more activity evolved around these new ideas (see ).…”