Colorful Carathéodory, Helly and Sierksma numbers of convexity spaces

Abstract

Since the appearance of the so-called colorful Carathéodory and Helly theorems in $\mathbb{R}^d$, many papers have been published with colorful generalizations of classical results in combinatorial geometry. The growing number of such publications is a good motivation to consider colorful analogs of known results in a more abstract setting. This is why we introduce in this note the colorful Carathéodory, Helly and Sierksma numbers of a convexity space. We study relationships between these new numbers and the classical (monochromatic) Carathéodory, Helly and exchange numbers of convexity spaces. Special attention is paid to the new numbers in product convexity and convex sum spaces.