标题： 不可避免复形的拓扑学 显示英文标题

标题： Topology of unavoidable complexes

摘要： 一个单纯复形 $K\subset 2^{[m]}$ 的划分数 $\pi(K)$ 是最小的整数 $\nu$，使得对于 $[m]$ 的任意划分 $A_1\uplus\ldots\uplus A_\nu = [m]$，至少有一个集合 $A_i$ 属于 $K$。 如果$\pi(K)\leq r$，则复杂度$K$是$r$-不可避免的。 We say that a complex $K$ is globally $r$-non-embeddable in $\mathbb{R}^d$ if for each continuous map $f: | K| \rightarrow \mathbb{R}^d$ there exist $r$ vertex disjoint faces $\sigma_1,\ldots, \sigma_r$ of $| K|$ such that $f(\sigma_1)\cap\ldots\cap f(\sigma_r)\neq\emptyset$. 受Tverberg-Van Kampen-Flores型问题的启发，我们证明了若干结果（定理3.6、3.9、4.6），这些结果将这两类复形的组合学与拓扑学联系起来。我们的一项中心观察（定理4.6），总结并扩展了G. Schild、B. Grünbaum及其他许多人的成果，即有趣的（全局）$r$-不可嵌入复形例子可以在$r$-不可避免复形的联结$K = K_1\ast\ldots\ast K_s$中找到。 显示英文摘要

摘要： The partition number $\pi(K)$ of a simplicial complex $K\subset 2^{[m]}$ is the minimum integer $\nu$ such that for each partition $A_1\uplus\ldots\uplus A_\nu = [m]$ of $[m]$ at least one of the sets $A_i$ is in $K$. A complex $K$ is $r$-unavoidable if $\pi(K)\leq r$. We say that a complex $K$ is globally $r$-non-embeddable in $\mathbb{R}^d$ if for each continuous map $f: | K| \rightarrow \mathbb{R}^d$ there exist $r$ vertex disjoint faces $\sigma_1,\ldots, \sigma_r$ of $| K|$ such that $f(\sigma_1)\cap\ldots\cap f(\sigma_r)\neq\emptyset$. Motivated by the problems of Tverberg-Van Kampen-Flores type we prove several results (Theorems 3.6, 3.9, 4.6) which link together the combinatorics and topology of these two classes of complexes. One of our central observations (Theorem 4.6), summarizing and extending results of G. Schild, B. Gr\"{u}nbaum and many others, is that interesting examples of (globally) $r$-non-embeddable complexes can be found among the joins $K = K_1\ast\ldots\ast K_s$ of $r$-unavoidable complexes.