Note on the Colored Tverberg Theorem

A Geometric Proof of the Colored Tverberg Theorem

The colored Tverberg theorem asserts that for every d and r there exists t = t(d, r) such that for every set C ⊂ R of cardinality (d + 1)t, partitioned into t-point subsets C1, C2, . . . , Cd+1 (which we think of as color classes; e.g., the points of C1 are red, the points of C2 blue, etc.), there exist r disjoint sets R1, R2, . . . , Rr ⊆ C that are rainbow, meaning that |Ri ∩ Cj| ≤ 1 for ever...

A tight colored Tverberg theorem for maps to manifolds

We prove that any continuous map of an N -dimensional simplex ∆N with colored vertices to a d-dimensional manifold M must map r points from disjoint rainbow faces of ∆N to the same point in M : For this we have to assume that N ≥ (r − 1)(d + 1), no r vertices of ∆N get the same color, and our proof needs that r is a prime. A face of ∆N is a rainbow face if all vertices have different colors. Th...

Optimal bounds for the colored Tverberg problem

We prove a “Tverberg type” multiple intersection theorem. It strengthens the prime case of the original Tverberg theorem from 1966, as well as the topological Tverberg theorem of Bárány et al. (1980), by adding color constraints. It also provides an improved bound for the (topological) colored Tverberg problem of Bárány & Larman (1992) that is tight in the prime case and asymptotically optimal ...

Notes on the topological Tverberg theorem

On the Number of Colored Birch and Tverberg Partitions

In 2009, Blagojević, Matschke & Ziegler established the first tight colored Tverberg theorem. We develop a colored version of our previous results (2008): Evenness and non-trivial lower bounds for the number of colored Tverberg partitions. Both properties follow from similar results on the number of colored Birch partitions.