Optimal bounds for a colorful Tverberg--Vrecica type problem

We prove the following optimal colorful Tverberg-Vrecica type transversal theorem: For prime r and for any k+1 colored collections of points C^l of size |C^l|=(r-1)(d-k+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 k-plane that meets all the convex hulls conv(F_j^l), under the assumption that r(d-k) 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 Borsuk-Ulam type theorem for (Z_p)^m-equivariant bundles that generalizes results of Volovikov (1996) and Zivaljevic (1999).Comment: Substantially revised version: new notation, improved results, additional references; 12 pages, 2 figure

О теоремах Р. Радо и Д. Уотсона

Theorems of R. Rado and of D. Watson are generalizations for theorems P. Kirchberger and C. Caratheodory. In the paper we will consider some generalizations and refinements for these theorems .Даётся некоторое обобщение теорем Р. Радо и Д. Уотсона, которые, в свою очередь, обобщают теоремы Каратеодори и Кирхбергера

A generalization of Gale's lemma

In this work, we present a generalization of Gale's lemma. Using this generalization, we introduce two combinatorial sharp lower bounds for ${\rm conid}({\rm B}_0(G))+1$ and ${\rm conid}({\rm B}(G))+2$, two famous topological lower bounds for the chromatic number of a graph $G$

On theorems of R. Rado and of D. Watson

Theorems of R. Rado and of D. Watson are generalizations for theorems P. Kirchberger and C. Caratheodory. In the paper we will consider some generalizations and refinements for these theorems