Large Convex sets in Difference sets

TL;DR

This paper constructs convex sets with large difference sets containing convex subsets and investigates the size of matchings with convex difference sets, establishing bounds and optimality.

Contribution

It provides a construction of convex sets with large convex difference subsets and determines the maximal size of matchings with convex difference sets, proving bounds are tight.

Findings

Constructed convex sets with difference sets containing convex subsets of size Omega(n^2)

Proved existence of matchings of size at least sqrt(n) with convex difference sets

Established that the lower bound on matching size is optimal up to a constant

Abstract

We give a construction of a convex set $A \subset \mathbb R$ with cardinality $n$ such that $A-A$ contains a convex subset with cardinality $\Omega (n^2)$. We also consider the following variant of this problem: given a convex set $A$, what is the size of the largest matching $M \subset A \times A$ such that the set \[ \{ a-b : (a,b) \in M \} \] is convex? We prove that there always exists such an $M$ with $|M| \geq \sqrt n$, and that this lower bound is best possible, up a multiplicative constant.