Explicit Ramsey graphs and Erdos distance problem over finite Euclidean and non-Euclidean spaces

TL;DR

This paper investigates the Erdős distance problem over finite Euclidean and non-Euclidean spaces using graphs related to these spaces, establishing their asymptotic Ramanujan properties to derive bounds and construct explicit Ramsey graphs.

Contribution

It introduces a novel approach using explicit graphs associated with finite spaces to improve bounds on the Erdős distance problem and construct explicit Ramsey graphs.

Findings

Graphs are asymptotically Ramanujan.

Derived new lower bounds with explicit constants.

Constructed explicit tough Ramsey graphs R(3,k).

Abstract

We study the Erdos distance problem over finite Euclidean and non-Euclidean spaces. Our main tools are graphs associated to finite Euclidean and non-Euclidean spaces that are considered in Bannai-Shimabukuro-Tanaka (2004, 2007). These graphs are shown to be asymptotically Ramanujan graphs. The advantage of using these graphs is twofold. First, we can derive new lower bounds on the Erdos distance problems with explicit constants. Second, we can construct many explicit tough Ramsey graphs R(3,k).