Note on the colored Tverberg theorem (Q1907116)

The colored Tverberg theorem proved by \textit{R. Živaljević} and \textit{S. Vrećica} [J. Comb. Theory, Ser. A 61, 309-318 (1992; Zbl 0782.52003)] and conjectured by \textit{I. Bárány, Z. Füredi} and \textit{L. Lovász} [Combinatorica 10, 175-183 (1990; Zbl 0718.52009)] states that: For any \(r,d > 1\) and large enough \(T\) and given disjoint \(T\)-element sets \(A_1,\dots, A_{d + 1}\) in \(\mathbb{R}^d\), one can find \(r\) disjoint transversals \(S_1, \dots, S_r\) of the sets \(A_1,\dots, A_{d + 1}\) such that the simplices spanned by \(S_1, \dots, S_r\) have a common point. The author provides a new proof of this result by verifying the following theorem: Let \(K\) be the simplicial complex of partial transversal of a set system of \(d + 1\) disjoint sets of cardinality \(2p -1\) each, \(p\) prime. Then for any map \(f : K \to \mathbb{R}^d\), there are \(p\) disjoint simplices of \(K\) such that the intersection of their images is not empty. From this the colored Tverberg theorem is easily deduced. The author very carefully introduces the topological tools needed for the proof. Basically, the statement is reduced to a result by \textit{A. Dold} [Contemp. Math. 19, 65-69 (1983; Zbl 0521.55002)] that says that for a prime \(p\) there is no \(\mathbb{Z}_p\)-map from a \(k\)-connected \(\mathbb{Z}_p\)-space into a \(\mathbb{Z}_p\)-space of dimension \(\leq k\).