Growth rates and critical exponents of classes of binary combinatorial geometries

Abstract:

We prove that a binary geometry of rank $n\;(n \geqslant 2)$ not containing $M({K_5})$ and ${F_7}$ (respectively, $M({K_5})$ and ${C_{10}}$) as a minor has at most $3n - 3$ (respectively, $4n - 5$) points. Here, $M({K_5})$ is the cycle geometry of the complete graph on five vertices, ${F_7}$ the Fano plane, and ${C_{10}}$ a certain rank $4$ ten-point geometry containing the dual Fano plane $F_7^{\ast }$ as a minor. Our technique is elementary and uses the notion of a bond graph. From these results, we deduce upper bounds on the critical exponents of these geometries.