The problem of deciding if a given triangulation of a sphere can be realized as the boundary sphere of a simplicial, convex polytope is known as the 'Simplicial Steinitz problem'. It is known by an indirect and non-constructive argument that a vast majority of Bier spheres are non-polytopal. Contrary to that, we demonstrate that the Bier spheres associated to threshold simplicial complexes are all polytopal. Moreover, we show that all Bier spheres are starshaped. We also establish a connection between Bier spheres and Kantorovich-Rubinstein polytopes by showing that the boundary sphere of the KR-polytope associated to a polygonal linkage (weighted cycle) is isomorphic to the Bier sphere of the associated simplicial complex of "short sets".[MPV] (with J. Melleray and F. Petrov) and to Kantorovich himself, see [K-R] where the Kantorovich-Rubinstein norm µ − ν KR is introduced.Bier spheres Bier(K), where K 2 [n] is an abstract simplicial complex, are combinatorially defined triangulations of the (n − 2)-dimensional sphere S n−2 with interesting combinatorial and topological properties, [Lo1, M03]. These spheres are known to be shellable [BPSZ, ČD]. Moreover it is known, by an indirect and non-effective counting argument, that the majority of these spheres are non-polytopal, in the sense that they do not admit a convex polytope realization, see [M03, Section 5.6]. They also provide one of the most elegant proofs of the Van Kampen-Flores theorem [M03] and serve as one of the main examples of "Alexander complexes" [JNPZ].Threshold complexes are ubiquitous in mathematics and arise, often in disguise and under different names, in areas as different as cooperative game theory (quota complexes and simple games) and algebraic topology of configuration spaces (polygonal linkages, complexes of short sets) [CoDe, Far, GaPa].The following theorem establishes a connection between the boundary ∂KR(d L ) of the Kantorovich-Rubinstein polytope of a weighted cycle, and the Bier sphere of the threshold complex of "short sets" of the associated polygonal linkage.

Conway and Lagarias observed that a triangular region T(m) in a hexagonal lattice admits a signed tiling by three-in-line polyominoes (tribones) if and only if m 2 {9d?1, 9d}d2N. We apply the theory of Gr?bner bases over integers to show that T(m) admits a signed tiling by n-in-line polyominoes (n-bones) if and only if m 2 {dn2 ? 1, dn2}d2N. Explicit description of the Gr?bner basis allows us to calculate the ?Gr?bner discrete volume? of a lattice region by applying the division algorithm to its ?Newton polynomial?. Among immediate consequences is a description of the tile homology group for the n-in-line polyomino. [Projekat Ministarstva nauke Republike Srbije, br. 174020 i br. 174034]

The partition number π(K) of a simplicial complex K ⊆ 2 [n] is the minimum integer k such that for each partition A 1 . .Motivated by the Van Kampen-Flores and Tverberg type results, and inspired by the 'constraint method' [BFZ], we study the combinatorics of r-unavoidable complexes. Emphasizing the interplay of ideas from combinatorial topology, linear programming and fractional graph theory, we explore and compare extremal properties of examples arising in topology (minimal triangulations) and combinatorics (hypergraph theory and Ramsey theory).

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