Abstract
In set theory without the Axiom of Choice (AC), we investigate the set-theoretic strength of Dilworth’s theorem for infinite posets with finite width, and its possible placement in the hierarchy of weak choice principles.
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References
Blass, A.: Ramsey's theorem in the hierarchy of choice principles. J. Symbolic Logic 42, 387–390 (1977)
P. J. Cameron, Combinatorics: Topics, Techniques, Algorithms, Cambridge University Press (Cambridge, UK, 1994).
de Bruijn, N.G., Erdős, P.: A colour problem for infinite graphs and a problem in the theory of relations. Indag. Math. 13, 371–373 (1951)
Dilworth, R.P.: A decomposition theorem for partially ordered sets. Ann. of Math. 51, 161–166 (1950)
Erné, M.: Prime Ideal Theorems and systems of finite character. Comment. Math. Univ. Carolinae 38, 513–536 (1997)
Galvin, F.: A proof of Dilworth's chain decomposition theorem. Amer. Math. Monthly 101, 352–353 (1994)
H. Herrlich, Axiom of Choice, Lecture Notes in Mathematics, Vol. 1876, Springer-Verlag (Berlin, Heidelberg, 2006).
Howard, P., Rubin, J.E.: Consequences of the Axiom of Choice, Mathematical Surveys and Monographs, vol. 59. American Mathematical Society (Providence, RI (1998)
Howard, P., Tachtsis, E.: On a variant of Rado's selection lemma and its equivalence with the Boolean prime ideal theorem. Arch. Math. Logic 53, 825–833 (2014)
T. J. Jech, The Axiom of Choice, Studies in Logic and the Foundations of Mathematics, Vol. 75, North-Holland Publishing Co. (Amsterdam, 1973)
Loeb, P.A.: A new proof of the Tychonoff theorem. Amer. Math. Monthly 72, 711–717 (1965)
L. Mirsky, Transversal Theory, Academic Press (New York and London, 1971)
Morillon, M.: Some consequences of Rado's selection lemma. Arch. Math. Logic 51, 739–749 (2012)
Mycielski, J.: Some remarks and problems on the coloring of infinite graphs and the theorem of Kuratowski. Acta Math. Acad. Sci. Hungar. 12, 125–129 (1961)
Perles, M.A.: A proof of Dilworth's decomposition theorem for partially ordered sets. Israel J. Math. 1, 105–107 (1963)
Perles, M.A.: On Dilworth's theorem in the infinite case. Israel J. Math. 1, 108–109 (1963)
Tachtsis, E.: On Ramsey's Theorem and the existence of infinite chains or infinite anti-chains in infinite posets. J. Symbolic Logic 81, 384–394 (2016)
E. Tachtsis, Łoś' Theorem and the Axiom of Choice, Math. Logic Quart. (to appear)
Tverberg, H.: On Dilworth's decomposition theorem for partially ordered sets. J. Combin. Theory 3, 305–306 (1967)
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I am very thankful to the anonymous referee for careful reading of the paper and for useful comments and suggestions which improved its quality.
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Tachtsis, E. Dilworth's decomposition theorem for posets in ZF. Acta Math. Hungar. 159, 603–617 (2019). https://doi.org/10.1007/s10474-019-00967-w
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DOI: https://doi.org/10.1007/s10474-019-00967-w