The Matroid Ramsey Number n(6,6)

Abstract

A tight upper bound on the number of elements in a connected matroid with fixed rank and largest cocircuit size is given. This upper bound is used to show that a connected matroid with at least thirteen elements contains either a circuit or a cocircuit with at least six elements. In the language of matroid Ramsey numbers, n(6, 6) = 13: this is the largest currently known matroid Ramsey number.

On the interplay between graphs and matroids

“…More generally, it follows from a result of Seymour [48] that the last inequality holds for all binary matroids that have no F * 7 -minor using e. In fact, it also holds for F * 7 , although it fails, for example, for AG (3,2). The last matroid is the matroid on the points of the three-dimensional affine space over GF (2) where a set of elements is independent in the matroid if the corresponding set of points is affinely independent.…”

“…Consider the following matrix A over GF (2), the field of two elements. Let E = {1, 2, ..., 8} and let C be the collection of minimal linearly dependent subsets of E. Again, (E, C) is a matroid.…”

“…For example, if we apply this operation six times to F − 7 , we obtain the uniform matroid U 3,7 . The Fano matroid is isomorphic to the vector matroid of the matrix whose columns consist of all non-zero vectors in the 3-dimensional vector space over GF (2). Hence the Fano matroid is binary where, in general, a matroid M is binary if M is isomorphic to the vector matroid of some matrix over the field GF (2).…”

“…To see that F 7 is non-graphic, it suffices to observe that there is no simple graph on 4 vertices with exactly 7 edges since the complete graph K 4 has only 6 edges. The matroid U 2,4 is non-binary because there is no 2 × 4 matrix A over GF (2) …”

“…The reader may hope that Seymour's decomposition theorem for regular matroids mentioned above would enable the graph result to be extended to regular matroids but this was not achieved either. Several authors [2,4,22,23,41,62,63] considered this problem and, in particular, Reid [41] proved several attractive results. Eventually, Bonin, McNulty and Reid [2] conjectured that the same bound holds for all matroids.…”