Some Small Circuit-Cocircuit Ramsey Numbers for Matroids

Ramsey numbers for matroids, which mimic properties of Ramsey numbers for graphs, have been denned as follows. Let k and l be positive integers. Then n ( k, l ) is the least positive integer n such that every connected matroid with n elements contains either a circuit with at least k elements or a cocircuit with at least l elements. We determine the largest known value of these numbers in the sense of maximizing both k and l . We also find extremal matroids with small circuits and cocircuits. Results on matroid connectivity, geometry, and extremal matroid theory are used here.