A recursive Lov\'asz theta number for simplex-avoiding sets

TL;DR

This paper introduces a recursive extension of the Lovász theta number to geometric hypergraphs, providing new bounds for independence ratios and applications in Euclidean Ramsey theory, including improved exponential bounds for simplexes.

Contribution

It develops a recursive Lovász theta number for geometric hypergraphs, offering new upper bounds and applications in Euclidean Ramsey theory.

Findings

Reproved that every k-simplex is exponentially Ramsey in the measurable setting

Improved bounds for the exponential base in Euclidean Ramsey theory

Provided a new upper bound for the independence ratio of geometric hypergraphs

Abstract

We recursively extend the Lov\'asz theta number to geometric hypergraphs on the unit sphere and on Euclidean space, obtaining an upper bound for the independence ratio of these hypergraphs. As an application we reprove a result in Euclidean Ramsey theory in the measurable setting, namely that every $k$-simplex is exponentially Ramsey, and we improve existing bounds for the base of the exponential.