LAUSR

The partition invariant $$\pi (K)$$ π ( K ) of a simplicial complex $$K\subseteq 2^{[m]}$$ K ⊆ 2 [ m ] is the minimum integer $$\nu $$ ν , such that for each partition $$A_1\uplus \cdots \uplus A_\nu = [m]$$ A 1 ⊎ ⋯ ⊎ A ν = [ m ] of [ m ], at least one of the sets $$A_i$$ A i is in K . A complex K is r -unavoidable if $$\pi (K)\le r$$ π ( K ) ≤ r . We say that a complex K is almost r -non-embeddable in $${\mathbb {R}}^d$$ R d if, for each continuous map $$f: \vert K\vert \rightarrow {\mathbb {R}}^d$$ f : | K | → R d , there exist r vertex disjoint faces $$\sigma _1,\cdots , \sigma _r$$ σ 1 , ⋯ , σ r of $$\vert K\vert $$ | K | , such that $$f(\sigma _1)\cap \cdots \cap f(\sigma _r)\ne \emptyset $$ f ( σ 1 ) ∩ ⋯ ∩ f ( σ r ) ≠ ∅ . One of our central observations (Theorem  2.1 ), summarizing and extending results of Schild et al. is that interesting examples of (almost) r -non-embeddable complexes can be found among the joins $$K = K_1*\cdots *K_s$$ K = K 1 ∗ ⋯ ∗ K s of r -unavoidable complexes.