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Some Small Circuit-Cocircuit Ramsey Numbers for Matroids

Published online by Cambridge University Press:  12 September 2008

Fair Barbour Hurst  and
Talmage James Reid
Fair Barbour Hurst
Affiliation:
Department of Mathematics, The University of Mississippi, University, MS 38677, U.S.A.
Talmage James Reid
Affiliation:
Department of Mathematics, The University of Mississippi, University, MS 38677, U.S.A.
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Abstract

Ramsey numbers for matroids, which mimic properties of Ramsey numbers for graphs, have been denned as follows. Let k and l be positive integers. Then n(k, l) is the least positive integer n such that every connected matroid with n elements contains either a circuit with at least k elements or a cocircuit with at least l elements. We determine the largest known value of these numbers in the sense of maximizing both k and l. We also find extremal matroids with small circuits and cocircuits. Results on matroid connectivity, geometry, and extremal matroid theory are used here.

Type
Research Article
Information
Combinatorics, Probability and Computing , Volume 4 , Issue 1 , March 1995 , pp. 67 - 80
Copyright
Copyright © Cambridge University Press 1995

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References

[1]Brylawski, T. H. (1971) A combinatorial model for series-parallel networks. Trans. Amer. Math. Soc. 154 1–22.CrossRefGoogle Scholar
[2]Chartrand, G. and Lesniak, L. (1986) Graphs and Digraphs, 2nd ed., Wadsworth.Google Scholar
[3]Cunningham, W. H. (1973) A combinatorial decomposition theory, Ph.D. thesis, University of Waterloo.Google Scholar
[4]Erdős, P. and Szekeres, G. (1935) A combinatorial problem in geometry. Compositio Math. 2 463–470.Google Scholar
[5]Greene, C. and Magnanti, T. (1975) Some abstract pivot algorithms. SI AM J. Appl. Math. 29 530–539.CrossRefGoogle Scholar
[6]Hurst, F. and Reid, T. J. (to appear) Ramsey numbers for cocircuits in matroids. Ars Combinatorica.Google Scholar
[7]Krogdahl, S. (1977) The dependence graph for bases in matroids. Discrete Math. 19 47–59.CrossRefGoogle Scholar
[8]Kung, J. P. S. (1993) Extremal matroid theory. In: Robertson, N. and Seymour, P. (eds.) Proceedings of the AMS-IMS-SIAM Joint Summer Research Conference on Graph Minors, University of Washington, Seattle, WA, 1991 Contemporary Math. 147 21–61.Google Scholar
[9]Nešetřil, J., Poljak, S. and Turzík, D. (1981) Amalgamation of matroids and its applications. J. Combin. Theory Ser. B 31 9–22.CrossRefGoogle Scholar
[10]Nešetřil, J., Poljak, S. and Turzík, D. (1985) Special amalgams and Ramsey matroids. In: Lovász, L. and Recski, A. (eds.) Matroid Theory, Colloq. Math. Soc. János Bolyai 40267–298.Google Scholar
[11]Oxley, J. G. (1981) On 3-connected matroids. Canad. J. Math. 33 20–27.CrossRefGoogle Scholar
[12]Oxley, J. G. (1992) Matroid Theory, Oxford University Press.Google Scholar
[13]Reid, T. J. (submitted) Ramsey numbers for matroids.Google Scholar
[14]Seymour, P. D. (1980) Decomposition of regular matroids. J. Combin. Theory Ser. B 28, 305–359.CrossRefGoogle Scholar
[15]Thomas, R. (private communication).Google Scholar
[16]Tutte, W. T. (1966) Connectivity in matroids. Canad. J. Math. 18 1301–1324.CrossRefGoogle Scholar
[17]Tuza, Z. (1987) On two intersecting set systems and k-continuous Boolean functions. Discrete Appl. Math. 16 183–185.CrossRefGoogle Scholar
[18]Welsh, D. J. A. (1976) Matroid Theory, Academic Press.Google Scholar