We recursively extend the Lovász 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.
- Pub Date:
- June 2021
- Mathematics - Combinatorics;
- Mathematics - Metric Geometry;
- (v2) 14 pages, 3 figures, background information on Euclidean Ramsey theory added