Skip to content
Accessible Unlicensed Requires Authentication Published by De Gruyter July 19, 2019

A dual Ramsey theorem for finite ordered oriented graphs

Dragan Mašulović and Bojana Pantić
From the journal Mathematica Slovaca

Abstract

In contrast to the abundance of “direct” Ramsey results for classes of finite structures (such as finite ordered graphs, finite ordered metric spaces and finite posets with a linear extension), in only a handful of cases we have a meaningful dual Ramsey result. In this paper we prove a dual Ramsey theorem for finite ordered oriented graphs. Instead of embeddings, which are crucial for “direct” Ramsey results, we consider a special class of surjective homomorphisms between finite ordered oriented graphs. Since the setting we are interested in involves both structures and morphisms, all our results are spelled out using the reinterpretation of the (dual) Ramsey property in the language of category theory.

  1. (Communicated by Peter Horák)

References

[1] Abramson, F. G.—Harrington, L. A.: Models without indiscernibles, J. Symbolic Logic 43 (1978), 572–600.Search in Google Scholar

[2] Aranda, A.—Bradley-Williams, D.—Hubička, J.—Karamanlis, M.—Kompatscher, M.—Konečný, M.—Pawliuk, M.: Ramsey expansions of metrically homogeneous graphs, arXiv:1707.02612.Search in Google Scholar

[3] Carlson, T. J.—Simpson, S. G.: A dual form of Ramsey’s theorem, Adv. Math. 53 (1984), 265–290.Search in Google Scholar

[4] Frankl, P.—Graham, R. L.—Rödl, V.: Induced restricted Ramsey theorems for spaces, J. Combin. Theory Ser. A 44 (1987), 120–128.Search in Google Scholar

[5] Graham, R. L.—Rothschild, B. L.: Ramsey’s theorem for n-parameter sets, Tran. Amer. Math. Soc. 159 (1971), 257–292.Search in Google Scholar

[6] Leeb, K.: The Categories of Combinatorics. Combinatorial Structures and their Applications, Gordon and Breach, New York, 1970.Search in Google Scholar

[7] Mašulović, D.: A dual Ramsey theorem for permutations, Electron. J. Combin. 24(3) (2017), #P3.39Search in Google Scholar

[8] Mašulović, D.: Dual Ramsey theorems for relational structures, Czechoslovak Math. J., to appear.Search in Google Scholar

[9] Mašulović, D.—Scow, L.: Categorical equivalence and the Ramsey property for finite powers of a primal algebra, Algebra Universalis 78 (2017), 159–179Search in Google Scholar

[10] Nešetřil, J.: Ramsey theory. In: Handbook of Combinatorics (R. L. Graham, M. Grötschel and L. Lovász, eds.), Vol. 2, MIT Press, Cambridge, MA, USA, 1995, pp. 1331–1403.Search in Google Scholar

[11] Nešetřil, J.—Rödl, V.: Partitions of finite relational and set systems, J. Combin. Theory Ser. A 22 (1977), 289–312.Search in Google Scholar

[12] Nešetřil, J.—Rödl, V.: Dual Ramsey type theorems. In: Proc. Eighth Winter School on Abstract Analysis (Z. Frolíik, ed.), Prague, 1980, pp. 121–123.Search in Google Scholar

[13] Nešetřil, J.—Rödl, V.: Ramsey classes of set systems, J. Combin. Theory Ser. A 34 (1983), 183–201.Search in Google Scholar

[14] Nešetřil, J.—Rödl, V.: The partite construction and Ramsey set systems, Discrete Math. 75 (1989), 327–334.Search in Google Scholar

[15] Prömel, H. J.: Induced partition properties of combinatorial cubes, J. Combin. Theory Ser. A 39 (1985), 177–208.Search in Google Scholar

[16] Prömel H. J.—Voigt, B.: Hereditary atributes of surjections and parameter sets, European J. Combin. 7 (1986), 161–170.Search in Google Scholar

[17] Ramsey, F. P.: On a problem of formal logic, Proc. London Math. Soc. 30 (1930), 264–286.Search in Google Scholar

[18] Sokić, M.: Ramsey properties of finite posets II, Order 29 (2012), 31–47.Search in Google Scholar

[19] Solecki, S.: A Ramsey theorem for structures with both relations and functions, J. Combin. Theory Ser. A 117 (2010), 704–714.Search in Google Scholar

[20] Solecki, S.: Dual Ramsey theorem for trees, arXiv:1502.04442.Search in Google Scholar

Received: 2018-03-07
Accepted: 2019-01-28
Published Online: 2019-07-19
Published in Print: 2019-08-27

© 2019 Mathematical Institute Slovak Academy of Sciences