We discuss dual Ramsey statements for several classes of finite relational structures (such as finite linearly ordered graphs, finite linearly ordered metric spaces and finite posets with a linear extension) and conclude the paper with another rendering of the Nešetřil-Rödl Theorem for relational structures. Instead of embeddings which are crucial for “direct” Ramsey results, for each class of structures under consideration we propose a special class of quotient maps and prove a dual Ramsey theorem in such a setting. Although our methods are based on reinterpreting the (dual) Ramsey property in the language of category theory, all our results are about classes of finite structures.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
F.G. Abramson, L. A. Harrington: Models without indiscernibles. J. Symb. Log. 43 (1978), 572–600.
J. Adámek, H. Herrlich, G. E. Strecker: Abstract and Concrete Categories: The Joy of Cats. Dover Books on Mathematics, Dover Publications, Mineola, 2009. zbl MR
P. Frankl, R. L. Graham, V. Rödl: Induced restricted Ramsey theorems for spaces. J. Comb. Theory, Ser. A 44 (1987), 120–128.
R. L. Graham, B. L. Rothschild: Ramsey’s theorem for n-parameter sets. Trans. Am. Math. Soc. 159 (1971), 257–292.
D. Mašulović: A dual Ramsey theorem for permutations. Electron. J. Comb. 24 (2017), Article ID P3.39, 12 pages. zbl MR
D. Mašulović: Pre-adjunctions and the Ramsey property. Eur. J. Comb. 70 (2018), 268–283.
D. Mašulović, L. Scow: Categorical equivalence and the Ramsey property for finite powers of a primal algebra. Algebra Univers. 78 (2017), 159–179.
J. Nešetřil: Ramsey theory. Handbook of Combinatorics, Vol. 2 (R. L. Graham et al., eds.). Elsevier, Amsterdam, 1995, pp. 1331–1403. zbl MR
J. Nešetřil: Metric spaces are Ramsey. Eur. J. Comb. 28 (2007), 457–468.
J. Nešetřil, V. Rödl: Partitions of finite relational and set systems. J. Comb. Theory, Ser. A 22 (1977), 289–312.
J. Nešetřil, V. Rödl: Dual Ramsey type theorems. Abstracta Eighth Winter School on Abstract Analysis, Mathematical Institute AS CR, Prague (Z. Frolík, ed.). 1980, pp. 121–123.
J. Nešetřil, V. Rödl: Ramsey classes of set systems. J. Comb. Theory, Ser. A 34 (1983), 183–201.
J. Nešetřil, V. Rödl: The partite construction and Ramsey set systems. Discrete Math. 75 (1989), 327–334.
H. J. Prömel: Induced partition properties of combinatorial cubes. J. Comb. Theory, Ser. A 39 (1985), 177–208.
H. J. Prömel, B. Voigt: Hereditary attributes of surjections and parameter sets. Eur. J. Comb. 7 (1986), 161–170.
H. J. Prömel, B. Voigt: A sparse Graham-Rothschild theorem. Trans. Am. Math. Soc. 309 (1988), 113–137.
F. P. Ramsey: On a problem of formal logic. Proc. Lond. Math. Soc. 30 (1930), 264–286.
M. Sokić: Ramsey properties of finite posets II. Order 29 (2012), 31–47.
S. Solecki: A Ramsey theorem for structures with both relations and functions. J. Comb. Theory, Ser. A 117 (2010), 704–714.
J.H. Spencer: Ramsey’s theorem for spaces. Trans. Am. Math. Soc. 249 (1979), 363–371.
The author gratefully acknowledges the support of the Grant No. 174019 of the Ministry of Education, Science and Technological Development of the Republic of Serbia.
About this article
Cite this article
Mašulović, D. On Dual Ramsey Theorems for Relational Structures. Czech Math J 70, 553–585 (2020). https://doi.org/10.21136/CMJ.2020.0408-18
- dual Ramsey property
- finite relational structure
- category theory