  Alexander $r$-tuples are introduced as a common generalization of pairs of
Alexander dual complexes (Alexander $2$-tuples) and $r$-unavoidable complexes
of Blagojevi\'{c}, Frick and Ziegler. The associated "Bier complexes" include
both the Bier spheres and "optimal multiple chessboard complexes" as
interesting, special cases. Our main result is Theorem 4.3 saying that (1) the
$r$-fold deleted join of Alexander $r$-tuple is a pure complex homotopy
equivalent to a wedge of spheres, and (2) the $r$-fold deleted join of a
collectively unavoidable $r$-tuple is $(n-r-1)$-connected. We also give a
complete classification (Theorem 5.1 and Corollary 5.2) of Alexander $r$-tuples
and Bier complexes.
