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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 $f(\\sigma_1)\\cap\\ldots\\cap f(\\sigma_r)\\neq\\emptyset$. 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