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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Constructing prime-field planar configurations

Author: Gary Gordon
Journal: Proc. Amer. Math. Soc. 91 (1984), 492-502
MSC: Primary 05B35; Secondary 51D20
MathSciNet review: 744655
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Abstract: An infinite class of planar configurations is constructed with distinct prime-field characteristic sets (i.e., configurations represented over a finite set of prime fields but over fields of no other characteristic). It is shown that if $ p$ is sufficiently large, then every subset of $ k$ primes between $ p$ and $ f(p,k)$ forms such a set (where $ f(p,k) = {2^{[(\sqrt p - A{k^{3/2}})/B{k^{3/2}}]}}$ for constants $ A$ and $ B$). In particular, for every positive integer $ k$, there exist infinitely many planar matroid configurations $ {C_{i,k}}$ with $ \left\vert {{\chi _{pf}}({C_{i,k}})} \right\vert = k$ (where $ {\chi _{pf}}(C)$ denotes the prime-field characteristic set of $ C$). We also give a result concerning cofinite prime-field characteristic sets.

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Keywords: Matroid configuration, characteristic set, prime-field characteristic set
Article copyright: © Copyright 1984 American Mathematical Society