We introduce and study Alexander r-tuples K = <K-i >(r)(i=1) of simplicial complexes, as a common generalization of pairs of Alexander dual complexes (Alexander 2-tuples) and r-unavoidable complexes of [3] and [11]. In the same vein, the Bier complexes, defined as the deleted joins K-Delta*of Alexander r-tuples, include both standard Bier spheres and optimal multiple chessboard complexes (Section 2.2) as interesting, special cases.

Our main results are Theorem 4.1 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 collective unavoidable r-tuple is (n - r - 1)-connected, and a classification theorem (Theorem 5.1 and Corollary 5.1) for Alexander r-tuples and Bier complexes.