The colored Tverberg's problem and complexes of injective functions (Q1200006)

Let \(t\), \(r\), \(d\) be positive integers and let \(C_ 1,\dots,C_{d+1}\) be a collection of disjoint sets in \(\mathbb{R}^ d\), called colors, each of cardinality at least \(t\). The ``colored Tverberg problem'' asks for the smallest value of \(t\), depending on \(r\) and \(d\), such that for every collection of colors there exist \(r\) disjoint, multicolored sets \(S_ i\subseteq\bigcup_ i C_ i\), \(i=1,\dots,r\), such that \(\bigcap_ i\text{conv}(S_ i)\neq\emptyset\). \(S\) is called multicolored if \(S\cap C_ i\neq\emptyset\) for all \(i=1,\dots,d+1\). The main theorem of this paper says that this smallest value is at most \(2r-1\) for every prime number \(r\). The proof uses a general version of the Borsuk-Ulam theorem for certain highly connected simplicial complexes of injective functions. This emphasizes a topological aspect of the colored Tverberg problem.