Dimensional colorful Helly theorems and topological variants

Abstract

We prove a dimensional strengthening of the colorful Helly theorem. Let ${\mathcal{C}}_1,\dots ,{\mathcal{C}}_{d+1}$ be finite nonempty families of convex sets in ${\mathbb{R}}^d$. If $C_1\cap \cdots \cap C_{d+1}\ne \varnothing$ for every choice of $C_i\in {\mathcal{C}}_i$, then ${\sum }_{i=1}^{d+1}\dim (\bigcap {\mathcal{C}}_i)\ge 0$, where $\dim \varnothing =-1$. We also obtain a dimensional strengthening of a theorem of Kim and Lew, in which the intersections are taken over unions of color classes. For simplicial complexes, we introduce the link-Leray dimension, defined as the Leray number of the link for a face and as $-1$ for a nonface. Using this invariant, we prove a matroidal extension of the dimensional colorful Helly theorem for $d$-Leray complexes. We further establish a corresponding strengthening of the topological Kim--Lew theorem in two ranges of parameters. As a special case, we recover the topological colorful Helly theorem of Kalai and Meshulam. Finally, we construct counterexamples to the unrestricted topological extension, even under a stronger local condition.