Cone and constrained colorful Carathéodory Theorems
Holmsen proved in 2016 a generalization of the classical colorful Caratheodory theorem in which a matroid imposes additional constraints on the desired colorful transversal. His approach also works in the more general setting of oriented matroids, rather than relying directly on convex hulls. In this paper, we extend these ideas in several directions. First, we study which colorful Caratheodory-type results remain valid when convex cones replace convex hulls, as well as analogous modifications in the oriented matroid setting. Second, we consider variants in which the additional constraint on the transversal is not encoded by a matroid. This leads to new extensions of the classical Tverberg theorem. Our approach is topological, following the methods of Holmsen, and Kalai and Meshulam, on which it builds. The key idea is to analyze homology groups of simplicial complexes that encode colorful Caratheodory-type phenomena, such as the support complex of an oriented matroid. In particular, one shows that these complexes are (near-)d-Leray. We extend this analysis by carrying out more detailed homology computations for these complexes, with the aim of enabling further and more refined applications of the method.