Branch-width and well-quasi-ordering in matroids and graphs. (Q1850591)

The authors prove that given a finite field \(F\) and an integer \(n\), each infinite collection of \(F\)-representable matroids with branch-width at most \(n\) has two members such that one is a minor of the other. They show that finiteness of \(F\) is crucial by showing that for each infinite field there exist infinite antichains of branch-width 3 that are representable over that field. The proof ideas are introduced by providing a direct proof of the \textit{N. Robertson} and \textit{P. D. Seymour} analog for graphs in [Graph minors. IV: Tree-width and well-quasi-ordering, J. Comb. Theory, Ser. B 48, No. 2, 227--254 (1990; Zbl 0719.05032)].