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Journal Article

# Constructing Prime-Field Planar Configurations

Gary Gordon
Proceedings of the American Mathematical Society
Vol. 91, No. 3 (Jul., 1984), pp. 492-502
DOI: 10.2307/2045328
Stable URL: http://www.jstor.org/stable/2045328
Page Count: 11

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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 $f(p, k) = 2^{\lbrack (\sqrt p - Ak^{3/2})/Bk^{3/2} \rbrack}$ for constants A and B). In particular, for every positive integer k, there exist infinitely many planar matroid configurations Ci, k with |χpf(Ci, k)| = k (where χpf(C) denotes the prime-field characteristic set of C). We also give a result concerning cofinite prime-field characteristic sets.

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