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].
@article{10_37236_7945,
author = {Zeev Dvir and Shay Moran},
title = {A {Sauer-Shelah-Perles} lemma for sumsets},
journal = {The electronic journal of combinatorics},
year = {2018},
volume = {25},
number = {4},
doi = {10.37236/7945},
zbl = {1400.05250},
url = {http://geodesic.mathdoc.fr/articles/10.37236/7945/}
}
TY - JOUR
AU - Zeev Dvir
AU - Shay Moran
TI - A Sauer-Shelah-Perles lemma for sumsets
JO - The electronic journal of combinatorics
PY - 2018
VL - 25
IS - 4
UR - http://geodesic.mathdoc.fr/articles/10.37236/7945/
DO - 10.37236/7945
ID - 10_37236_7945
ER -
%0 Journal Article
%A Zeev Dvir
%A Shay Moran
%T A Sauer-Shelah-Perles lemma for sumsets
%J The electronic journal of combinatorics
%D 2018
%V 25
%N 4
%U http://geodesic.mathdoc.fr/articles/10.37236/7945/
%R 10.37236/7945
%F 10_37236_7945
Zeev Dvir; Shay Moran. A Sauer-Shelah-Perles lemma for sumsets. The electronic journal of combinatorics, Tome 25 (2018) no. 4. doi: 10.37236/7945
