On $k$-connected Sets in Matroids

Source abstract

In 1966, Tutte defined $k$-connected matroids for each $k \ge 2$. Robertson and Seymour built on Tutte's idea to define tangles as a way to identify highly connected parts of a matroid. Motivated by tangles, a set $Z$ in a matroid $M$ has been defined to be $3$-connected if, for each $j$ in $\{1,2\}$, there is no $j$-separation $(X,Y)$ of $M$ such that each of $|Z \cap X|$ and $|Z \cap Y|$ is at least $j$. This paper solves a problem of Whittle for $k= 3$ by proving that a minor-minimal matroid having $Z$ as a $3$-connected set has at most $3|Z| - 4$ elements. We also identify the matroids for which this bound is attained.