A problem of P. Seymour on nonbinary matroids
It is well-known that a matroid is non-binary if and only if it has a \(U_{2,4}\)-minor. This paper looks at the problem of relating sets of elements in a non-binary matroid M to \(U_{2,4}\)-minors of M. \textit{R. E. Bixby} [Discrete Math. 8, 139-145 (1974; Zbl 0279.05027)] proved that, when \(k=2\), if M is k-connected and X is a (k-1)-element subset of M, then M has a \(U_{2,4}\)-minor whose ground set contains X. The same result was proved for \(k=3\) by \textit{P. D. Seymour} [Combinatorica 1, 387- 394 (1981; Zbl 0489.05020)] who also asked whether it remains true for \(k=4\). This paper answers this question negatively. The author's example is a 15-element rank-7 matroid \(M_ 1\) that is obtained from a binary matroid \(M_ 2\) with the same rank and ground set by relaxing a circuit- hyperplane of the latter. This means that the independent sets of \(M_ 1\) are all the independent sets of \(M_ 2\) together with a single circuit-hyperplane of \(M_ 2\). Another counterexample to the case \(k=4\) of this result, a 12-element rank-6 matroid, has been independently found by \textit{C. Coullard} [Counterexamples to conjectures on 4-connected matroids, Combinatorica (to appear)]. After establishing that his example has the properties claimed, the author concludes with two conjectures, the first of which has been proved by the reviewer [Combinatorica 5, 343- 345 (1985)].
scientific article; zbMATH DE number 3952784