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Martin’s axiom and partitions. (English) Zbl 0643.03033

This paper brings together MA, the structure of ccc posets, and questions of partition calculus. The main tool is the following: Definition. A partition \([X]^{\kappa}=P_ 0\cup P_ 1\) (resp. \([X]^{<\kappa}=P_ 0\cup P_ 1)\) is ccc-destructible iff there is a ccc partial order forcing “there is an uncountable \(Y\subset X\) so that \([Y]^{\kappa}\) (resp. \([Y]^{<\kappa})\) is a subset of \(P_ 0.''\) Thus “every uncountable ccc poset has an uncountable centered subset” is equivalent to: “if \([X]^{<\omega}=P_ 0\cup P_ 1\) is ccc-destructible then X has an uncountable 0-homogeneous subset.”
The main results are:
Theorem 1. \(MA_{\kappa}\) iff every ccc poset of size \(\kappa\) is \(\sigma\)-centered.
Theorem 2. \(MA_{\omega_ 1}\) iff every uncountable ccc poset has an uncountable centered subset.
Theorem 3. There is a \(\sigma\)-linked poset of size t without centered subsets of size t. (Here t \(=\) the minimal size of a tower in \({\mathcal P}(\omega).)\)
Theorem 4. There is a \(\sigma\)-linked poset of size p which is not \(\sigma\)-centered. (Here p \(=\) the minimal size of a family in \({\mathcal P}(\omega)/fin\) with no lower bound.)
Theorem 5. There is a ccc non-separable compact Hausdorff space of size \(c.\)
Theorem 6. Under any of the following assumptions there is a ccc- destructible partition of \([\omega_ 1]^ 3\) with no uncountable 0- homogeneous subsets: (a) there is non-\(\sigma\)-linked ccc poset of size \(\omega_ 1\); (b) \({\mathfrak c}<2^{\omega_ 1}\); (c) there is a non- special Aronszajn tree.
The reader should be warned that the paper leaves more to the reader than is usual even for these authors. Most striking are theorem 6(a) above, which is stated in the introduction and never mentioned again, and theorem 6(c) above, stated as a corollary of a more technical theorem not mentioned here, without proof. The interested reader can piece together the proof of 6(a) from corollary 2.6 p. 405 and the unproved remark in italics in the middle of p. 398; 6(c) follows from 6(a) once you observe that the poset forcing an Aronszajn tree T to be special is ccc but fails to be \(\sigma\)-linked if T is not special.
Reviewer: J.Roitman

MSC:

03E05 Other combinatorial set theory
03E50 Continuum hypothesis and Martin’s axiom
03E35 Consistency and independence results
54A35 Consistency and independence results in general topology
06A06 Partial orders, general
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References:

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