Balanced Signings and the Chromatic Number of Oriented Matroids

We consider the problem of reorienting an oriented matroid so that all its cocircuits are 'as balanced as possible in ratio'. It is well known that any oriented matroid having no coloops has a totally cyclic reorientation, a reorientation in which every signed cocircuit B = {B + , B − } satisfies B + , B − = ∅. We show that, for some reorientation, every signed cocircuit satisfies 1/f (r) ≤ |B + |/|B − | ≤ f (r), where f (r) ≤ 14 r 2 ln(r), and r is the rank of the oriented matroid. In geometry, this problem corresponds to bounding the discrepancies (in ratio) that occur among the Radon partitions of a dependent set of vectors. For graphs, this result corresponds to bounding the chromatic number of a connected graph by a function of its Betti number (corank) |E| − |V | + 1.