Local Properties in Colored Graphs, Distinct Distances, and Difference Sets

TL;DR

This paper explores how local properties in colored graphs, geometric point sets, and additive sets can imply global combinatorial bounds, introducing a novel non-algebraic energy method to derive these results.

Contribution

It introduces a new non-algebraic energy technique to establish bounds in extremal combinatorics problems involving local-to-global property implications.

Findings

Bounds for a Ramsey problem with colored graphs are tight in various cases.

New bounds for point sets with many distinct distances in small subsets.

Improved estimates for difference sets in additive combinatorics.

Abstract

We study Extremal Combinatorics problems where local properties are used to derive global properties. That is, we consider a given configuration where every small piece of the configuration satisfies some restriction, and use this local property to derive global properties of the entire configuration. We study one such Ramsey problem of Erd\H{o}s and Shelah, where the configurations are complete graphs with colored edges and every small induced subgraph contains many distinct colors. Our bounds for this Ramsey problem show that the known probabilistic construction is tight in various cases. We study one Discrete Geometry variant, also by Erd\H{o}s, where we have a set of points in the plane such that every small subset spans many distinct distances. Finally, we consider an Additive Combinatorics problem, where we are given sets of real numbers such that every small subset has a large difference set. We derive new bounds for all of the above problems. Our proof technique is based on introducing an non-algebraic variant of additive energies. This abstract energy variant is based on edge colors in graphs.