On nonbinary $3$-connected matroids
JAMES G. OXLEY · Transactions of the American Mathematical Society · 1987
It is well known that a matroid is binary if and only if it has no minor isomorphic to ${U_{2,4}}$, the $4$-point line. Extending this result, Bixby proved that every element in a nonbinary connected matroid is in a ${U_{2,4}}$-minor. The result was further extended by Seymour who showed that every pair of elements in a nonbinary $3$-connected matroid is in a ${U_{2,4}}$-minor. This paper extends Seymour’s theorem by proving that if $\left \{ {x, y, z} \right \}$ is contained in a nonbinary $3$-connected matroid $M$, then either $M$ has a ${U_{2,4}}$-minor using $\left \{ {x, y, z} \right \}$, or $M$ has a minor isomorphic to the rank-$3$ whirl that uses $\left \{ {x, y, z} \right \}$ as its rim or its spokes.