Partition numbers. (English) Zbl 0891.03019

For a partial order \(P\) let \({\mathfrak a}(P)\) denote the least cardinal such that there is no absolute partition of \(P\) of that size. In Section 1 the author investigates suborderings of \(({\mathcal P} (\omega)/ \text{fin})^\omega\). So let \(P_f\) and \(P_c\) denote the suborderings of \(({\mathcal P} (\omega)/ \text{fin})^\omega\) which consist of filtered elements and chains, respectively. The author shows that \({\mathfrak a}(P_f)= {\mathfrak a} (P_c)\), and that both numbers are greater than or equal to \({\mathfrak p}\) (the minimal cardinality of a filter on \({\mathcal P} (\omega)/ \text{fin}\) which has no lower bound).
For \(X\) an uncountable Polish space, let \({\mathfrak a} (X)\) be the minimal size of an uncountable partition of \(X\) into closed sets. In Section 2 it is shown that the dominating number is less than or equal to \({\mathfrak a} (X)\). Using Miller forcing he shows that the dominating number can be less than \({\mathfrak a} (X)\).
In Section 3 the author shows that the partition number for Mathias forcing is greater than or equal to the bounding number.
In Section 4 he deals with the lattice of partitions of \(\omega\). Let \((\omega)^\omega\) be the set of infinite partitions of \(\omega\). The author shows that it is consistent with ZFC that the distributivity number of \((\omega)^\omega\) is less than the distributivity number of \({\mathcal P} (\omega)/ \text{fin}\).
Reviewer: M.Weese (Potsdam)


03E05 Other combinatorial set theory
03E35 Consistency and independence results
06A07 Combinatorics of partially ordered sets
54H05 Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)
Full Text: DOI


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