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• W2891647548 abstract "The partition number $pi(K)$ of a simplicial complex $Ksubset 2^{[m]}$ is the minimum integer $nu$ such that for each partition $A_1uplusldotsuplus A_nu = [m]$ 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 $f(sigma_1)capldotscap f(sigma_r)neqemptyset$. 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. Grunbaum and many others, is that interesting examples of (globally) $r$-non-embeddable complexes can be found among the joins $K = K_1astldotsast K_s$ of $r$-unavoidable complexes." @default.
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• W2891647548 date "2016-03-28" @default.
• W2891647548 modified "2023-09-27" @default.
• W2891647548 title "Topology of unavoidable complexes" @default.
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