Oriented matroids with few mutations
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- Richter-Gebert, J. Discrete Comput Geom (1993) 10: 251. doi:10.1007/BF02573980
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This paper defines a “connected sum” operation on oriented matroids of the same rank. This construction is used for three different applications in rank 4. First it provides nonrealizable pseudoplane arrangements with a low number of simplicial regions. This contrasts the case of realizable hyperplane arrangements: by a classical theorem of Shannon every arrangement ofn projective planes in ℝPd-1 contains at leastn simplicial regions and every plane is adjacent to at leastd simplicial regions , . We construct a class of uniform pseudoarrangements of 4n pseudoplanes in ℝP3 with only 3n+1 simplicial regions. Furthermore, we construct an arrangement of 20 pseudoplanes where one plane is not adjacent to any simplicial region.
Finally we disprove the “strong-map conjecture” of Las Vergnas . We describe an arrangement of 12 pseudoplanes containing two points that cannot be simultaneously contained in an extending hyperplane.