Notre Dame Journal of Formal Logic

Ramsey Theory for Countable Binary Homogeneous Structures

Jean A. Larson

Abstract

Countable homogeneous relational structures have been studied by many people. One area of focus is the Ramsey theory of such structures. After a review of background material, a partition theorem of Laflamme, Sauer, and Vuksanovic for countable homogeneous binary relational structures is discussed with a focus on the size of the set of unavoidable colors.

Article information

Source
Notre Dame J. Formal Logic, Volume 46, Number 3 (2005), 335-352.

Dates
First available in Project Euclid: 30 August 2005

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1125409332

Digital Object Identifier
doi:10.1305/ndjfl/1125409332

Mathematical Reviews number (MathSciNet)
MR2162104

Zentralblatt MATH identifier
1095.03035

Subjects
Primary: 03E02: Partition relations 03C15: Denumerable structures
Secondary: 05A15: Exact enumeration problems, generating functions [See also 33Cxx, 33Dxx]

Keywords
Ramsey theory partition relation relational structure canonical partition enumeration random graph Rado graph

Citation

Larson, Jean A. Ramsey Theory for Countable Binary Homogeneous Structures. Notre Dame J. Formal Logic 46 (2005), no. 3, 335--352. doi:10.1305/ndjfl/1125409332. https://projecteuclid.org/euclid.ndjfl/1125409332


Export citation

References

  • [1] Baldwin, J. T., "Rank and homogeneous structures", pp. 215--33 in Tits Buildings and the Model Theory of Groups (Würzburg, 2000), vol. 291, Cambridge University Press, Cambridge, 2002.
  • [2] Baldwin, J. T., and K. Holland, "Constructing $\omega$"-stable structures: Model completeness, Annals of Pure and Applied Logic, vol. 125 (2004), pp. 159--72.
  • [3] Bonato, A., P. Cameron, and D. Delić, "Tournaments and orders with the pigeonhole property", Canadian Mathematical Bulletin, vol. 43 (2000), pp. 397--405.
  • [4] Cameron, P. J., "The random graph", pp. 333--51 in The Mathematics of Paul Erdős, II, vol. 14, Springer, Berlin, 1997.
  • [5] Cherlin, G., "The classification of countable homogeneous directed graphs and countable homogeneous $n$"-tournaments, Memoirs of the American Mathematical Society, vol. 131 (1998).
  • [6] Cherlin, G., "Sporadic homogeneous structures", pp. 15--48 in The Gelfand Mathematical Seminars, 1996--1999, Birkhäuser Boston, Boston, 2000.
  • [7] Cherlin, G., and N. Shi, "Forbidden subgraphs and forbidden substructures", The Journal of Symbolic Logic, vol. 66 (2001), pp. 1342--52.
  • [8] Devlin, D. C., Some Partition Theorems and Ultrafilters on $\omega$, Ph.D. thesis, Dartmouth College, Hanover, 1979.
  • [9] Droste, M., and D. Kuske, "On random relational structures", Journal of Combinatorial Theory. Series A, vol. 102 (2003), pp. 241--54.
  • [10] Džamonja, M., and S. Shelah, "Universal graphs at the successor of a singular cardinal", The Journal of Symbolic Logic, vol. 68 (2003), pp. 366--88.
  • [11] Erdős, P., A. Hajnal, and L. Pósa, "Strong embeddings of graphs into colored graphs", pp. 109--30 in Infinite and Finite Sets, vol. 10, dedicated to P. Erdős on his 60th birthday, North-Holland, Amsterdam, 1975.
  • [12] Erdős, P., and A. Rényi, "Asymmetric graphs", Acta Mathematica Academiae Scientiarum Hungaricae, vol. 14 (1963), pp. 295--315.
  • [13] Erdős, P., and R. Rado, "A combinatorial theorem", Journal of the London Mathematical Society. Second Series, vol. 25 (1950), pp. 249--55.
  • [14] Evans, D. M., "$\aleph\sb 0$"-categorical structures with a predimension, Annals of Pure and Applied Logic, vol. 116 (2002), pp. 157--86.
  • [15] Fraï"ssé, R., Theory of Relations, revised edition, vol. 145 of Studies in Logic and the Foundations of Mathematics, North-Holland Publishing Co., Amsterdam, 2000.
  • [16] Furkhen, E. G., "Languages with added quantifier `there exist at least $\aleph_\alpha$'", pp. 121--31 in The Theory of Models, edited by erseeditorsnames J. V. Addison, L. A. Henkin, and A. Tarski, North Holland Publishing Company, Amsterdam, 1965.
  • [17] Gilchrist, M. and S. Shelah, ``Identities on cardinals less than $\aleph_\omega$,'' The Journal of Symbolic Logic, vol. 61 (1996), pp. 780--87.
  • [18] Graham, R. L., D. E. Knuth, and O. Patashnik, Matematyka Konkretna, 2d edition, Wydawnictwo Naukowe PWN, Warsaw, 1998.
  • [19] Henson, C. W., "A family of countable homogeneous graphs", Pacific Journal of Mathematics, vol. 38 (1971), pp. 69--83.
  • [20] Hrushovski, E., "A new strongly minimal set", Annals of Pure and Applied Logic, vol. 62 (1993), pp. 147--66.
  • [21] Kechris, A. S., V. G. Pestov, and S. Todorčević, "Fraï"ssé limits, Ramsey theory, and topological dynamics of automorphism groups, vol. 15 (2005), pp. 109--89, preprint posted at arXiv:math.LO/0305241.
  • [22] Kojman, M., and S. Shelah, "Nonexistence of universal orders in many cardinals", The Journal of Symbolic Logic, vol. 57 (1992), pp. 875--91.
  • [23] Lachlan, A. H., "Countable homogeneous tournaments", Transactions of the American Mathematical Society, vol. 284 (1984), pp. 431--61.
  • [24] Lachlan, A. H., "On countable stable structures which are homogeneous for a finite relational language", Israel Journal of Mathematics, vol. 49 (1984), pp. 69--153.
  • [25] Lachlan, A. H., and R. E. Woodrow, "Countable ultrahomogeneous undirected graphs", Transactions of the American Mathematical Society, vol. 262 (1980), pp. 51--94.
  • [26] Laflamme, C., N. W. Sauer, and V. Vuksanovic, "Canonical partitions of universal structures", preprint posted at arXiv:math.LO/0305241, 2003.
  • [27] Larson, J. A., "Counting canonical partitions in the random graph", preprint, 2004.
  • [28] Milliken, K. R., "A Ramsey theorem for trees", Journal of Combinatorial Theory. Series A, vol. 26 (1979), pp. 215--37.
  • [29] Rado, R., "Universal graphs and universal functions", Acta Arithmetica, vol. 9 (1964), pp. 331--40.
  • [30] Sauer, N. W., "Coloring subgraphs of the Rado graph", forthcoming in Combinatorica.
  • [31] Schmerl, J. H., "Countable homogeneous partially ordered sets", Algebra Universalis, vol. 9 (1979), pp. 317--21.
  • [32] Shelah, S., "Two cardinal compactness", Israel Journal of Mathematics, vol. 9 (1971), pp. 193--98.
  • [33] Shelah, S., "Independence results", The Journal of Symbolic Logic, vol. 45 (1980), pp. 563--73.
  • [34] Stanley, R. P., Enumerative Combinatorics. Vol. 2, vol. 62 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 1999.
  • [35] Vuksanovic, V., "A combinatorial proof of a partition relation for $[\mathbbQ]^n$", \emProceedings of the American Mathematical Society, vol. 130 (2002), pp. 2857--64.
  • [36] Vuksanovic, V., "Infinite partitions of random graphs", preprint, 2003.