The universal partition theorem for oriented matroids

Oriented matroids and Ky Fan's theorem

L. Lovász has shown in [9] that Sperner’s combinatorial lemma admits a generalization involving a matroid defined on the set of vertices of the associated triangulation. We prove that Ky Fan’s theorem admits an oriented matroid generalization of similar nature (Theorem 3.1). Classical Ky Fan’s theorem is obtained as a corollary if the underlying oriented matroid is chosen to be the alternating ...

A Topological Representation Theorem for tropical oriented matroids

Tropical oriented matroids were defined by Ardila and Develin in 2007. They are a tropical analogue of classical oriented matroids in the sense that they encode the properties of the types of points in an arrangement of tropical hyperplanes – in much the same way as the covectors of (classical) oriented matroids describe the types in arrangements of linear hyperplanes. Not every oriented matroi...

Menger's Theorem for Matroids

Menge r's Theore m asse rt s t hat if x and yare ve rti ces of a graph wh ic h a re not joined b y an edge and if it takes at least k ot he r vert ices to separate x and y, the n x and y can be JOined by k di s tinc t arcs "' the graph whic h have o nl y th eir e nd-vertices in common . . . Mu c h of the pape r is take n up by the de terminati on of an equiva le nt s tate me nt In volving th e ...

Fachbereich 3 Mathematik Oriented Matroids Oriented Matroids

A chain theorem for matroids

Tutte’s Wheels-and-Whirls Theorem proves that if M is a 3-connected matroid other than a wheel or a whirl, then M has a 3-connected minor N such that |E(M)| − |E(N)| = 1. Geelen and Whittle extended this theorem by showing that when M is sequentially 4-connected, the minor N can also be guaranteed to be sequentially 4connected, that is, for every 3-separation (X, Y ) of N , the set E(N) can be ...