On forbidden minors for 𝐺𝐹(3)

Kahn, Jeff; Seymour, Paul

doi:10.1090/S0002-9939-1988-0921013-1

On forbidden minors for $\textrm {GF}(3)$
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by Jeff Kahn and Paul Seymour PDF

Proc. Amer. Math. Soc. 102 (1988), 437-440 Request permission

Abstract:

A new, surprisingly simple proof is given of the finiteness of the set of matroids minor-minimally not representable over $GF(3)$. It is, in fact, proved that every such matroid has rank or corank at most 3.

References

Robert E. Bixby, On Reid’s characterization of the ternary matroids, J. Combin. Theory Ser. B 26 (1979), no. 2, 174–204. MR 532587, DOI 10.1016/0095-8956(79)90056-X

Uniquely representable combinatorial geometries

Jeff Kahn, A geometric approach to forbidden minors for $\textrm {GF}(3)$, J. Combin. Theory Ser. A 37 (1984), no. 1, 1–12. MR 749507, DOI 10.1016/0097-3165(84)90014-1
William M. Kantor, Envelopes of geometric lattices, J. Combinatorial Theory Ser. A 18 (1975), 12–26. MR 357164, DOI 10.1016/0097-3165(75)90062-x
Gian-Carlo Rota, Combinatorial theory, old and new, Actes du Congrès International des Mathématiciens (Nice, 1970) Gauthier-Villars, Paris, 1971, pp. 229–233. MR 0505646
P. D. Seymour, Matroid representation over $\textrm {GF}(3)$, J. Combin. Theory Ser. B 26 (1979), no. 2, 159–173. MR 532586, DOI 10.1016/0095-8956(79)90055-8
K. Truemper, Alpha-balanced graphs and matrices and $\textrm {GF}(3)$-representability of matroids, J. Combin. Theory Ser. B 32 (1982), no. 2, 112–139. MR 657681, DOI 10.1016/0095-8956(82)90028-4
W. T. Tutte, A homotopy theorem for matroids. I, II, Trans. Amer. Math. Soc. 88 (1958), 144–174. MR 101526, DOI 10.1090/S0002-9947-1958-0101526-0
D. J. A. Welsh, Matroid theory, L. M. S. Monographs, No. 8, Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York, 1976. MR 0427112