Applications of the Canonical Ramsey Theorem to Geometry

TL;DR

This paper applies the canonical Ramsey theorem to geometric problems, establishing bounds on the size of point subsets with distinct distances and areas in Euclidean spaces, using novel techniques.

Contribution

Introduces new geometric bounds based on canonical Ramsey theorem variants, improving understanding of point configurations with distinct distances and areas.

Findings

Subset with all distances distinct is at least Omega(n^{1/6d})

Subset with all areas distinct in 2D is at least Omega((log log n)^{1/186})

Results extend to countable point sets in R^d

Abstract

Let P be a set of n points in R^d. How big is the largest subset X of P such that all of the distances determined between pairs are different? We show that X is at at least Omega(n^{1/6d}) This is not the best known; however the technique is new. Assume that no three of the original points are collinear. How big is the largest subset X of P such that all of the areas determined by elements of all triples are different? We show that, if d=2 then X is at least Omega((log log n)^{1/186}) and if d=3 then X is at least Omega((log log n)^{1/396}). We also obtain results for countable sets of points in R^d. All of our proofs use variants of the canonical Ramsey theorem and some geometric lemmas.