Excluding the Cycle Geometries of the Kuratowski Graphs from Binary Geometries
Joseph P. S. Kung · Proceedings of the London Mathematical Society · 1987
We prove that a binary combinatorial geometry (or simple matroid) of rank n not containing M(K5) (respectively, M(K3,3)) as a minor contains at most 8n (respectively, 10n) points. Here, M(K5) is the cycle geometry of the complete graph K5 and M(K3,3) is the cycle geometry of the complete bipartite graph K3,3. These bounds are probably not tight. The proof uses the notion of a bond graph. Using these results, we obtain upper bounds on the critical exponents of these geometries.