Constructing primefield planar configurations
Author:
Gary Gordon
Journal:
Proc. Amer. Math. Soc. 91 (1984), 492502
MSC:
Primary 05B35; Secondary 51D20
MathSciNet review:
744655
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Abstract: An infinite class of planar configurations is constructed with distinct primefield 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 is sufficiently large, then every subset of primes between and forms such a set (where for constants and ). In particular, for every positive integer , there exist infinitely many planar matroid configurations with (where denotes the primefield characteristic set of ). We also give a result concerning cofinite primefield characteristic sets.
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 T. Brylawski, Finite primefield characteristic sets for planar configurations, Linear Algebra Appl. 46 (1982), 155176. MR 664703 (83h:05029)
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 T. Brylawski and D. Kelly, Matroids and combinatorial geometries, Lecture Notes Ser., University of North Carolina, Chapel Hill, N. C., 1980. MR 573268 (81f:05053)
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 T. Brylawski and D. Lucas, Uniquely representable combinatorial geometries, Proceedings of the International Colloquium on Combinatorial Theory, Rome, 1976, pp. 83104.
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 G. Gordon, Representations of matroids over prime fields, Ph.D. Thesis, University of North Carolina, Chapel Hill, N. C., 1983.
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 G. Hardy and E. Wright, An introduction to the theory of numbers, Oxford Univ. Press, London, 1938, 1945, 1954, 1960, 1979. MR 568909 (81i:10002)
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 A. W. Ingleton, Representation of matroids, Combinatorial Mathematics and its Applications (D. J. A. Welsh, ed.), Academic Press, New York, 1971, pp. 149167. MR 0278974 (43:4700)
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 J. Kahn, Characteristic sets of matroids, J. Loncon Math. Soc. (2) 26 (1982), 207217. MR 675165 (84j:05044)
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 R. Rado, Note on independence functions, Proc. London Math. Soc. (3) 7 (1957), 300320. MR 0088459 (19:522b)
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 R. Reid, Obstructions to representations of combinatorial geometries (unpublished; appears as Appendix in [2]).
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 W. Tutte, Lectures on matroids, J. Res. Nat. Bur. Standards 69B (1965), 147. MR 0179781 (31:4023)
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 P. Vamos, A necessary and sufficient condition for a matroid to be linear, Matroid Conf. (Brest, 1970). MR 0349447 (50:1941)
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 S. S. Wagstaff, Jr., Infinite matroids, Trans. Amer. Math. Soc. 175 (1973), 141153. MR 0398867 (53:2718)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939198407446551
PII:
S 00029939(1984)07446551
Keywords:
Matroid configuration,
characteristic set,
primefield characteristic set
Article copyright:
© Copyright 1984
American Mathematical Society
