Optimal bounds for a colorful TverbergVrecica type problem
Abstract
We prove the following optimal colorful TverbergVrecica type transversal theorem: For prime r and for any k+1 colored collections of points C^l of size C^l=(r1)(dk+1)+1 in R^d, where each C^l is a union of subsets (color classes) C_i^l of size smaller than r, l=0,...,k, there are partition of the collections C^l into colorful sets F_1^l,...,F_r^l such that there is a kplane that meets all the convex hulls conv(F_j^l), under the assumption that r(dk) is even or k=0. Along the proof we obtain three results of independent interest: We present two alternative proofs for the special case k=0 (our optimal colored Tverberg theorem (2009)), calculate the cohomological index for joins of chessboard complexes, and establish a new BorsukUlam type theorem for (Z_p)^mequivariant bundles that generalizes results of Volovikov (1996) and Zivaljevic (1999).
 Publication:

arXiv eprints
 Pub Date:
 November 2009
 arXiv:
 arXiv:0911.2692
 Bibcode:
 2009arXiv0911.2692B
 Keywords:

 Mathematics  Algebraic Topology;
 Mathematics  Combinatorics;
 52A35;
 55S35
 EPrint:
 Substantially revised version: new notation, improved results, additional references