Ample completions of oriented matroids and complexes of uniform oriented matroids

This paper considers completions of tope graphs of complexes of oriented matroids (COMs) to ample partial cubes of the same Vapnik-Chervonenkis dimension (VC-dimension). We show that these exist for oriented matroids (OMs) and complexes of uniform oriented matroids (CUOMs). This implies that tope graphs of OMs and CUOMs satisfy the sample compression conjecture one of the central open questions of learning theory. We conjecture that the tope graph of every COM can be completed to an ample partial cube without increasing the VC-dimension.