×

A Ramsey theorem for partial orders with linear extensions. (English) Zbl 1348.05210

Summary: We prove a Ramsey theorem for finite sets equipped with a partial order and a fixed number of linear orders extending the partial order. This is a common generalization of two recent Ramsey theorems due to M. Sokić [J. Comb. Theory, Ser. A 132, 142–171 (2015; Zbl 1307.05220)]. As a bonus, our proof gives new arguments for these two results.

MSC:

05D10 Ramsey theory

Citations:

Zbl 1307.05220
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Bodirsky, M., New Ramsey classes from old, Electron. J. Combin., 21, 2 (2014), paper 2.22 · Zbl 1300.05311
[2] Bodirsky, M., Ramsey classes: examples and constructions, (Surveys in Combinatorics 2015. Surveys in Combinatorics 2015, London Mathematical Society Lecture Note Series, vol. 424 (2015), Cambridge University Press), 1-48 · Zbl 1352.05183
[3] Böttcher, J.; Foniok, J., Ramsey properties of permutations, Electron. J. Combin., 20, 1 (2013), paper 2 · Zbl 1267.05284
[4] Fouché, W. L., Symmetry and the Ramsey degree of posets, Discrete Math., 167/168, 309-315 (1997) · Zbl 0873.05076
[5] Graham, R. L.; Rothschild, B. L.; Spencer, J. H., Ramsey Theory (1990), John Wiley & Sons · Zbl 0705.05061
[6] Kechris, A. S.; Pestov, V. G.; Todorcevic, S., Fraïssé limits, Ramsey theory, and topological dynamics of automorphism groups, Geom. Funct. Anal., 15, 106-189 (2005) · Zbl 1084.54014
[7] Nešetřil, J., Ramsey classes and homogeneous structures, Combin. Probab. Comput., 14, 171-189 (2005) · Zbl 1059.05103
[8] Nešetřil, J.; Rödl, V., Partitions of finite relational and set systems, J. Combin. Theory Ser. A, 22, 289-312 (1977) · Zbl 0361.05017
[9] Nguyen Van Thé, L., A survey on structural Ramsey theory and topological dynamics with the Kechris-Pestov-Todorcevic correspondence in mind, Zb. Rad., 17, 25, 189-207 (2015), Selected topics in combinatorial analysis · Zbl 1461.05232
[10] Paoli, M.; Trotter, W. T.; Walker, J. W., Graphs and orders in Ramsey theory and in dimension theory, (Graphs and Order (1985), Reidel), 351-394 · Zbl 0566.05045
[11] Sokić, M., Ramsey properties of finite posets, Order, 29, 1-30 (2012) · Zbl 1254.03066
[12] Sokić, M., Ramsey property, ultrametric spaces, finite posets, and universal minimal flows, Israel J. Math., 194, 609-640 (2013) · Zbl 1267.05286
[13] Sokić, M., Directed graphs and boron trees, J. Combin. Theory Ser. A, 132, 142-171 (2015) · Zbl 1307.05220
[14] Solecki, S., A Ramsey theorem for structures with both relations and functions, J. Combin. Theory, Ser. A, 117, 704-714 (2010) · Zbl 1247.05256
[15] Solecki, S., Abstract approach to finite Ramsey theory and a self-dual Ramsey theorem, Adv. Math., 248, 1156-1198 (2013) · Zbl 1283.05176
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.