Shallow Packings, Semialgebraic Set Systems, Macbeath Regions, and Polynomial Partitioning

Given a set system (X, R) such that every pair of sets in R have large symmetric dif-ference, the Shallow Packing Lemma gives an upper bound on |R| as a function of the shallow-cell complexity of R. In this paper, we first present a matching lower bound. Then we give our main theorem, an application of the Shallow Packing Lemma: given a semialgebraic set system (X, R) with shallow-cell complexity ϕ(·, ·) and a parameter ɛ > 0, there exists a collection, called an ɛ-Mnet, consisting of (Formula Presented) subsets of X, each of size (Formula Presented), such that any R ∈ R with | R| ≥ ɛ| X| contains at least one set in this collection. We observe that as an immediate corollary an alternate proof of the optimal ɛ-net bound follows.