Multiple chessboard complexes and the colored Tverberg problem

TL;DR

This paper investigates the connectivity and shellability of multiple chessboard complexes, providing sharp bounds that advance the understanding of Tverberg-type problems involving multiple points of the same color.

Contribution

It introduces new sharp connectivity bounds for multiple chessboard complexes, extending their application to Tverberg-type problems with multiple points of the same color.

Findings

Established sharp connectivity bounds for multiple chessboard complexes.

Provided foundational results for Tverberg-van Kampen-Flores type theorems.

Extended the theoretical framework for colored Tverberg problems.

Abstract

Following D.B. Karaguezian, V. Reiner, and M.L. Wachs (Matching Complexes, Bounded Degree Graph Complexes, and Weight Spaces of $GL$-Complexes, Journal of Algebra 2001) we study the connectivity degree and shellability of multiple chessboard complexes. Our central new results (Theorems 3.2 and 4.4) provide sharp connectivity bounds relevant to applications in Tverberg type problems where multiple points of the same color are permitted. These results also provide a foundation for the new results of Tverberg-van Kampen-Flores type, as announced in arXiv:1502.05290 [math.CO].