Two elements of a matroid are clones if the map that interchanges the two elements and fixes all other elements is an automorphism. Clones are important in the study of matroid representability. We give results on the number and size of clone sets in representable matroids.

Smith conjectured in 1979 that two distinct cycles in a k-connected graph meet in at least k vertices when k >= 2. Matroid extensions of this conjecture are considered. We also give a result that characterizes the connected binary matroids with two different circuit sizes.

Finally, we consider conjectures of Wu on deletable and contractible elements in 4-connected matroids. This is the result of work with many others such as Cotwright, Lemos, McMurray, Robbins, Sheppardson, Wei, Wu, and Zhou.