Simplicial complexes are among the central objects of study in topological and geometric combinatorics and their role in this subject can be compared to the role of manifolds in differential geometry and topology. For illustration, the “simple games” (Alexander self-dual complexes) arise in the description of the moduli spaces of planar polygonal linkages (arXiv:1209.3241v5), while the class of so called “symmetrized, multiple chessboard complexes” plays a central role in the proof of a general Van Kampen-Flores type theorem (arXiv:1502.05290). Topology and combinatorics of so called r-unavoidable complexes has been recently in the focus of wider mathematical audience after the breakthrough related to the Topological Tverberg conjecture, see AMS Notices (August 2016) report by I. Barany, P.V.M. Blagojevic, and G. Ziegler. Our objective is to develop (in part as a continuation of https://arxiv.org/abs/1607.07157 ) a unified approach to “r-unavoidable”, “chessboard complexes”, Alexander complexes, Bier complexes, and other related classes of simplicial complexes which play a distinct role in applications to discrete geometry, applied and computational algebraic topology, and cooperative game theory. ​