  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].
