Abraham, Uri; Rubin, Matatyahu; Shelah, Saharon On the consistency of some partition theorems for continuous colorings, and the structure of \(\aleph _ 1\)-dense real order types. (English) Zbl 0585.03019 Ann. Pure Appl. Logic 29, 123-206 (1985). The authors describe and apply various methods of constructing c.c.c. posets. Of these the club method is probably best-known: if one wants to build an object of size \(\omega_ 1\) with certain properties, then very often the obvious poset of finite approximations is not c.c.c.; the club method first chooses a club set C in \(\omega_ 1\), one can then take the poset of finite approximations which are separated by C, i.e. between any two elements of the approximation there is an element of C. If C is chosen careful enough then this poset will be c.c.c. Usually the club method requires CH; the authors also show how to remove CH in many constructions. This opens the way to proving statements like ”all \(\omega_ 1\)-dense subsets of \({\mathbb{R}}\) are isomorphic” consistent with \(2^{\omega}>\omega_ 1\). The CH-requirement prevented this at first. The paper contains much more than is described here and is recommended reading for anyone who wants to prove consistency results about sets of size \(\omega_ 1\). Reviewer: K.P.Hart Cited in 5 ReviewsCited in 66 Documents MSC: 03E05 Other combinatorial set theory 03E35 Consistency and independence results 03E50 Continuum hypothesis and Martin’s axiom Keywords:c.c.c. forcing; countable chain condition; c.c.c. posets; club method PDFBibTeX XMLCite \textit{U. Abraham} et al., Ann. Pure Appl. Logic 29, 123--206 (1985; Zbl 0585.03019) Full Text: DOI References: [1] Avraham, U.; Shelah, S., Martin’s axiom does not imply that every two \(ℵ_1\)-dense sets of reals are isomorphic, Israel J. Math., 38, 161-176 (1981) · Zbl 0457.03048 [2] Baumgartner, J. E., All \(ℵ_1\)-dense sets of reals can be isomorphic, Fund. Math., 79, 101-106 (1973) · Zbl 0274.02037 [3] Baumgartner, J. E., Chains and antichains in \(P(ω)\), J. Symbolic Logic, 45, 85-92 (1980) · Zbl 0437.03027 [4] Blass, A., A partition theorem for perfect sets, Proc. Amer. Math. Soc., 82, 271-277 (1981) · Zbl 0472.03038 [5] Bonnet, R.; Pouzet, M., Linear extensions of ordered sets, (Rival, I., Ordered Sets (1982), Reidel: Reidel Dordrecht) · Zbl 0499.06002 [6] Galvin, F.; Shelah, S., Some counterexamples in the partition calculus, J. Comb. Theory (A), 15, 167-174 (1973) · Zbl 0267.04006 [7] Magidor, M.; Malitz, J., Compact extensions of L(Q), Annals Math. Logic, 11, 217-261 (1977) · Zbl 0356.02012 [8] Sierpiński, W., Hypothèse du Continu (1956), Chelsea, New York · Zbl 0075.00903 [9] Solway, R. W.; Tennenbaum, S., Iterated Cohen extensions and Suslin’s problem, Ann. Math., 94, 201-245 (1971) · Zbl 0244.02023 [10] Kunen, K.; Tall, F. D., Between Martin’s axiom and Souslin hypothesis, Fund. Math., 102, 173-181 (1979) · Zbl 0415.03040 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.