A recursive Lovász theta number for simplex-avoiding sets

Abstract

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.

arXiv comment(v2) 14 pages, 3 figures, background information on Euclidean Ramsey theory added

References in corpus (9)

• New upper bounds on sphere packings I
• Progression-free sets in Z_4^n are exponentially small
• Upper bounds for packings of spheres of several radii
• Fourier analysis, linear programming, and densities of distance avoiding sets in R^n
• Lower bounds for measurable chromatic numbers
• Spectral bounds for the independence ratio and the chromatic number of an operator
• Monochromatic Equilateral Triangles in the Unit Distance Graph
• Complete positivity and distance-avoiding sets
• Computing upper bounds for the packing density of congruent copies of a convex body