  Chessboard complexes and their generalizations, as objects, and Discrete
Morse theory, as a tool, are presented as a unifying theme linking different
areas of geometry, topology, algebra and combinatorics. Edmonds and Fulkerson
bottleneck (minmax) theorem is proved and interpreted as a result about a
critical point of a discrete Morse function on the Bier sphere of an associated
simplicial complex $K$. We illustrate the use of "standard discrete Morse
functions" on generalized chessboard complexes by proving a connectivity result
for chessboard complexes with multiplicities. Applications include new
Tverberg-Van Kampen-Flores type results for $j$-wise disjoint partitions of a
simplex.
