On matroids of branch-width three.
Gian-Carlo Rota conjectured that the set of excluded minors for the class of matroids (combinatorial geometries) that are representable over a fixed finite field is finite [Actes Congr. internat. Math. 1970, 3, 229--233 (1971; Zbl 0362.05044)]. Some partial results are known. For all positive integers \(k\), the class \(\mathcal{B}_k\) of matroids of branch-width at most \(k\) is minor-closed. The first interesting class is \(\mathcal{B}_3\), containing infinite antichains of matroids and is thus not well-quasi-ordered under the minor order. In this paper the authors prove Rota'conjecture for \(\mathcal{B}_3\) (in the cases \(\mathcal{B}_1\) and \(\mathcal{B}_2\) the conjecture is trivial).
scientific article; zbMATH DE number 1843849