Abstract
A big Ramsey spectrum of a countable chain (i.e. strict linear order) C is a sequence of big Ramsey degrees of finite chains computed in C. In this paper we consider big Ramsey spectra of countable scattered chains. We prove that countable scattered chains of infinite Hausdorff rank do not have finite big Ramsey spectra, and that countable scattered chains of finite Hausdorff rank with bounded finite sums have finite big Ramsey spectra. Since big Ramsey spectra of all non-scattered countable chains are finite by results of Galvin, Laver and Devlin, in order to complete the characterization of countable chains with finite big Ramsey spectra (or degrees) one still has to resolve the remaining case of countable scattered chains of finite Hausdorff rank whose finite sums are not bounded.
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References
Devlin, D.: Some Partition theorems and Ultrafilters on ω Ph.D. Thesis, Dartmouth College (1979)
Erdős, P., Hajnal, A.: On a classification of denumerable order types and an application to the partition calculus. Fundam. Math. 51, 117–129 (1962)
Erdős, P., Hajnal, A.: Unsolved and solved problems in set theory. In: Proceedings of the Tarski Symposium (Proc. Sympos. Pure Math., Vol. XXV, Univ. California, Berkeley, Calif., 1971) Vol. 25, 269–287. Amer. Math. Soc. Providence, R.I (1974)
Galvin, F.: Partition theorems for the real line. Notices Amer. Math. Soc. 15, 660 (1968)
Galvin, F.: Errata to Partition theorems for the real line. Notices Amer. Math. Soc. 16, 1095 (1969)
Hausdorff, F.: Grundzüge einer Theorie der Geordnete Mengen. Math. Ann. 65, 435–505 (1908)
Kechris, A. S., Pestov, V. G., Todorčević, S.: Fraïssé limits, Ramsey theory and topological dynamics of automorphism groups. GAFA Geometric and Functional Analysis 15, 106–189 (2005)
Laver, R.: On fraïssé’s order type conjecture. Ann. Math. 93, 89–111 (1971)
Laver, R.: An order type decomposition theorem. Annals of mathematics. Second Series 98, 96–119 (1973)
Mašulović, D., Šobot, B.: Countable ordinals and big Ramsey degrees. Combinatorica 41, 425–446 (2021)
Ramsey, F. P.: On a problem of formal logic. Proc. London Math. Soc. 30, 264–286 (1930)
Rosenstein, J. G.: Linear Orderings. Academic Press (1982)
Todorčević, S.: Oscillations of sets of integers. Adv. Appl. Math. 20, 220–252 (1998)
Williams, N. H.: Combinatorial set theory. North-Holland (1977)
Zucker, A.: Big Ramsey Groups degrees and topological dynamics. Geometry and Dynamics 235–276 (2018)
Acknowledgements
The author would like to thank F. Galvin and S. Todorčević for their many insightful observations during the preparation of [10] which paved the road for this paper.
The author would also like to thank the two anonymous referees for the careful reading of the paper and many constructive comments that brought much needed clarity to the last two sections of the paper.
The author gratefully acknowledges the financial support of the Ministry of Education, Science and Technological Development of the Republic of Serbia (Grant No. 451-03-9/2021-14/200125).
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Mašulović, D. Big Ramsey Spectra of Countable Chains. Order 40, 237–256 (2023). https://doi.org/10.1007/s11083-022-09608-4
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DOI: https://doi.org/10.1007/s11083-022-09608-4