Equitable partitions of continuum

W. Holland

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We give some necessary and sufficient conditions for transitive $l$ -permutation groups to be $2$ -transitive. We also discuss primitive components and give necessary and sufficient conditions for transitive $l$ -permutation groups to be normal-valued.

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J. Truss (1978)

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On special partitions of Dedekind- and Russell-sets

Horst Herrlich, Paul Howard, Eleftherios Tachtsis (2012)

Commentationes Mathematicae Universitatis Carolinae

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A Russell set is a set which can be written as the union of a countable pairwise disjoint set of pairs no infinite subset of which has a choice function and a Russell cardinal is the cardinal number of a Russell set. We show that if a Russell cardinal $a$ has a ternary partition (see Section 1, Definition 2) then the Russell cardinal $a + 2$ fails to have such a partition. In fact, we prove that if a ZF-model contains a Russell set, then it contains Russell sets with ternary partitions as well...