On the number of triangles in 3-connected matroids

An element ee of a 3-connected matroid MM is said to be contractible provided that M/eM/e is 3-connected. In this paper, we show that a 3-connected matroid MM with exactly kk contractible elements has at least max{r∗(M)+6−2k4,|E(M)|+6−3k5} triangles. For each kk, we construct an infinite family of matroids that attain this bound. New sharp bounds for the number of triads of a minimally 3-connected matroid are obtained as a consequence of our main result.