The cocircuit graph of an oriented matroid is the 1-skeleton of the cellular decomposition induced by the Topological Representation Theorem due to Folkman & Lawrence (1978). In this paper we exhibit...

Abstract. There is a one-to-one correspondence between geometric lattices and the intersection lattices of arrangements of homotopy spheres. When the arrangements are essential and fully partitioned, Zaslavsky’s enumeration of the cells of the arrangement still holds. An application of the theory shows that all minimal cellular resolutions of matroid Steiner ideals are bounded subcomplexes of homotopy sphere arrangements of the given matroid. As a result the Betti numbers of the ideal are computed and seen to be equivalent to Stanley’s formula in the special case of face ideals of independence complexes of matroids. 1.

This paper regards an optimization problem which is an oriented matroid analogue of the graph chromatic number. There are several ways in which a `chromatic number' might be defined for more general matroids. One such formulation, introduced by Goddyn, Tarsi and Zhang [7], depends only on the sign patterns of (signed) circuits (or cocircuits). The result is a natural invariant of an oriented matroid. The invariant can be viewed as a `discrepancy in ratio' of a pseudo-hyperplane arrangement, and thus should be of interest to geometers. The main theorem answers a question raised in [7]. We first state the result and some consequences, using a minimal set of definitions. Detailed definitions appear in Section 2. It is convenient to view an oriented matroid to be a matroid in which every circuit C (and cocircuit B) has been partitioned C = C C - , (and B = B subject to a standard orthogonality condition. We regard each bipartition as an unordered pair }, where one of the parts may be empty. For E(O), the reorientation I of is the new oriented matroid obtained from by repartitioning each circuit C (and cocircuit B) according to the rules } ## {C C), C - C)} and } ## {B B), B - B)}

An arrangement of oriented pseudohyperplanes in affine d-space defines on its set X of pseudohyperplanes a set system (or range space) (X, R), R ⊆ 2 X of VCdimension d in a natural way: to every cell c in the arrangement assign the subset of pseudohyperplanes having c on their positive side, and let R be the collection of all these subsets. We investigate and characterize the range spaces corresponding to simple arrangements of pseudohyperplanes in this way; such range spaces are called pseudogeometric, and they have the property that the cardinality of R is maximum for the given VC-dimension. In general, such range spaces are called maximum, and we show that the number of ranges R ∈ R for which also X −R ∈ R, determines whether a maximum range space is pseudogeometric. Two other characterizations go via a simple duality concept and ‘small ’ subspaces. The correspondence to arrangements is obtained indirectly via a new characterization of uniform oriented matroids: a range space (X, R) naturally corresponds to a uniform oriented matroid of rank |X | − d if and only if its VC-dimension is d, R ∈ R implies X − R ∈ R and |R | is maximum under these conditions.

A recent progress on the complete enumeration of oriented matroids enables us to generate all combinatorial types of small point configurations and hyperplane arrangements in general dimension, including degenerate ones. This extends a number of former works which concentrated on the non-degenerate case and are usually limited to dimension 2 or 3. Our initial study on the complete list for small cases has shown its potential in resolving geometric conjectures. 1

this paper we explore the relationship between combinatorial vector bundles and real vector bundles. As a consequence of our results we get theorems relating the topology of the combinatorial Grassmannians to that of their real analogs. The theory of oriented matroids gives a combinatorial abstraction of linear algebra; a k-dimensional subspace of R

Separoids ---a natural generalization of oriented matroids--- are symetric relations 2 F defined on pairs of disjoint subsets which are closed, as filters, by the natural partial order D); they encode the separation structure of the families convex sets as follows: a pair of disjoint subsets A, B is a Radon partition, denoted as A+B, i# their convex hulls intersect; otherwise, we say that they are separated ---hence the name of the structure. We will show here that every separoid S can be represented (realized) with a family of convex sets ; that is, given an abstract (combinatorial) separoid S we will construct a family of convex sets whose separation structure (whose Radon partitions) are exactly those of the separoid S.

We consider the cocircuit graph GM of an oriented matroid M, which is the 1-skeletonof the cell complex formed by the span of the cocircuits of M. As a result of Cordovil, Fukuda, and Guedes de Oliveira, the isomorphism class of M is not determined by GM, butit is determined if M is uniform and the vertices in G M are paired if they are associatedto negative cocircuits; furthermore the reorientation class of an oriented matroid M with

This paper shows that it is NP-complete to algorithmically decide whether there exists a chirotope which extends a given partial chirotope. While Crippen and Havel ([3]) gave an algorithm to compute all possible extensions of a given partial chirotope running in exponential time, Gärtner et al. ([6]) discussed a special class of partial chirotopes which turned out to be always extendable.

Abstract. Combinatorial vector bundles, or matroid bundles, areacombinatorial analog to real vector bundles. Combinatorial objects called oriented matroids play the role of real vector spaces. This combinatorial analogy is remarkably strong, and has led to combinatorial results in topology and bundle-theoretic proofs in combinatorics. This paper surveys recent results on matroid bundles, and describes a canonical functor from real vector bundles to matroid bundles. 1.