维度彩色Helly定理及拓扑变体
Dimensional colorful Helly theorems and topological variants
- Moscow Institute of Physics and Technology（莫斯科物理技术学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文证明了彩色Helly定理的维度加强及其拓扑变体，引入链-Leray维数并建立拟阵扩展，同时构造反例表明无限制拓扑扩展不成立。
AI中文摘要：
我们证明了彩色Helly定理的一个维度加强版本。设$\mathcal{C}_1,\dots,\mathcal{C}_{d+1}$为$\mathbb{R}^d$中有限非空凸集族。若对每个选择$C_i\in\mathcal{C}_i$都有$C_1\cap\cdots\cap C_{d+1}\neq\varnothing$，则$\sum_{i=1}^{d+1}\dim(\bigcap\mathcal{C}_i)\geq0$，其中$\dim\varnothing=-1$。我们还获得了Kim和Lew定理的一个维度加强，其中交集取在颜色类的并集上。对于单纯复形，我们引入了链-Leray维数，定义为面的链的Leray数，对于非面则为$-1$。利用这个不变量，我们证明了$d$-Leray复形的维度彩色Helly定理的一个拟阵扩展。我们进一步在两组参数范围内建立了对应的拓扑Kim--Lew定理的加强。作为特例，我们恢复了Kalai和Meshulam的拓扑彩色Helly定理。最后，我们构造了无限制拓扑扩展的反例，即使在更强的局部条件下也是如此。
英文摘要：
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 $C_1\cap\cdots\cap C_{d+1}\neq\varnothing$ for every choice of $C_i\in\mathcal{C}_i$, then $\sum_{i=1}^{d+1}\dim(\bigcap\mathcal{C}_i)\geq0$, 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.
