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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Growth rates and critical exponents of classes of binary combinatorial geometries

Author: Joseph P. S. Kung
Journal: Trans. Amer. Math. Soc. 293 (1986), 837-859
MSC: Primary 05B35; Secondary 51D20
MathSciNet review: 816330
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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.

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Keywords: Matroid theory, minor-closed classes, growth rates, critical exponents, bond graphs
Article copyright: © Copyright 1986 American Mathematical Society