2017/02/27 by Pavel Paták, Paták, Pavel
Mathematics · #05B35 #51D20 #Combinatorics (math.CO) #FOS: Mathematics #math.CO #msc:05B35 #msc:51D20
paper · pdf · doi:10.48550/arxiv.1702.08170
arxiv created 2019/09/19 · arxiv updated 2019/09/20
In this paper we show a variant of colorful Tverberg's theorem which is valid in any matroid: Let S be a sequence of non-loops in a matroid M of finite rank m with closure operator cl. Suppose that S is colored in such a way that the first color does not appear more than r-times and each other color appears at most (r-1)-times. Then S can be partitioned into r rainbow subsequences S1,…, Sr such that cl ∅\subsetneq cl S1⊆ cl S2⊆ … ⊆ cl Sr. In particular, ∅≠ \bigcapi=1r cl Si. A subsequence is called rainbow if it contains each color at most once. The conclusion of our theorem is weaker than the conclusion of the original Tverberg's theorem in \mathbb Rd, which states that \bigcap conv Si≠ ∅, whereas we only claim that \bigcap aff Si≠ ∅. On the other hand, our theorem strengthens the Tverberg's theorem in several other ways: 1) it is applicable to any matroid (whereas Tverberg's theorem can only be used in \mathbb Rd), 2) instead of \bigcap cl Si≠ ∅ we have the stronger condition cl ∅\subsetneq cl S1⊆ cl S2⊆ … ⊆ cl Sr, and 3) we add a color constraints that are even stronger than the color constraints in the colorful version of Tverberg's theorem. Recently, the author together with Goaoc, Mabillard, Patáková, Tancer and Wagner used the first property and applied the non-colorful version of this theorem to homology groups with GF(p) coefficients to obtain several non-embeddability results, for details we refer to arXiv:1610.09063.