Growth Rates and Critical Exponents of Classes of Binary

We prove that a binary geometry of rank n {n > 2) not containing M(Ky and F-, (respectively, M(K5) and C10) as a minor has at most 3/i 3 (respectively, 4« 5) points. Here, M(K5) is the cycle geometry of the complete graph on five vertices, F-, the Fano plane, and C10 a certain rank 4 ten-point geometry containing the dual Fano plane F* 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.