Erd\H{o}s-Szekeres theorem for $k$-flats

TL;DR

This paper generalizes the Erdős-Szekeres theorem, originally about points, to higher-dimensional $k$-flats in Euclidean space, exploring combinatorial properties of flat configurations.

Contribution

It introduces a novel extension of the Erdős-Szekeres theorem to $k$-flats in ${ m f R}^d$, broadening the scope of classical combinatorial geometry.

Findings

Extended Erdős-Szekeres theorem to $k$-flats

Established bounds for flat configurations in Euclidean space

Provided new combinatorial insights into geometric arrangements

Abstract

We extend the famous Erd\H{o}s-Szekeres theorem to $k$-flats in ${\mathbb{R}^d}$