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...

We show a necessary and sufficient condition for any ordinal number to be Polish space. also prove that each countable space, there exists is an upper bound the first component of Cantor-Bendixson characteristic every compact subset aforementioned In addition, uncountable nonzero natural number, we existence this space such its equals previous pair numbers. Finally, determine cardinality partit...

The q-state Potts model can be defined on an arbitrary finite graph, and its partition function encodes much important information about that graph, including its chromatic polynomial, flow polynomial and reliability polynomial. The complex zeros of the Potts partition function are of interest both to statistical mechanicians and to combinatorists. I give a pedagogical introduction to all these...

Let σ be a finite relational signature and T a set of finite complete relational structures of signature σ and HT the countable homogeneous relational structure of signature σ which does not embed any of the structures in T . In the case that σ consists of at most binary relations and T is finite the vertex partition behaviour of HT is completely analysed; in the sense that it is shown that a c...

Recently Lewis Bowen introduced a notion of entropy for measure-preserving actions of a countable sofic group on a standard probability space admitting a generating partition with finite entropy. By applying an operator algebra perspective we develop a more general approach to sofic entropy which produces both measure and topological dynamical invariants. We establish the variational principle ...

There exists a family fB g <!1 of sets of countable ordinals such that (1) maxB = , (2) if 2 B then B B , (3) if and is a limit ordinal then B \ is not in the ideal generated by the B , < , and by the bounded subsets of , (4) there is a partition fAng1n=0 of !1 such that for every and every n; B \An is nite.

Assuming large cardinals we produce a forcing extension of V which preserves cardinals, does not add reals, and makes the set of points of countable V cofinality in κ nonstationary. Continuing to force further, we obtain an extension in which the set of points of countable V cofinality in ν is nonstationary for every regular ν ≥ κ. Finally we show that our large cardinal assumption is optimal. ...

There exists a family {Bα}α<ω1 of sets of countable ordinals such that (1) maxBα = α, (2) if α ∈ Bβ then Bα ⊆ Bβ , (3) if λ ≤ α and λ is a limit ordinal then Bα ∩ λ is not in the ideal generated by the Bβ , β < α, and by the bounded subsets of λ, (4) there is a partition {An}∞n=0 of ω1 such that for every α and every n, Bα∩An is finite.

There exists a family {Bα}α<ω1 of sets of countable ordinals such that (1) maxBα = α, (2) if α ∈ Bβ then Bα ⊆ Bβ , (3) if λ ≤ α and λ is a limit ordinal then Bα ∩ λ is not in the ideal generated by the Bβ , β < α, and by the bounded subsets of λ, (4) there is a partition {An}n=0 of ω1 such that for every α and every n, Bα∩An is finite.

It is well known that if G is a countable amenable group and G y (Y ,ν) factors onto G y (X ,μ), then the entropy of the first action must be at least the entropy of the second action. In particular, if G y (X ,μ) has infinite entropy, then the action G y (Y ,ν) does not admit any finite generating partition. On the other hand, we prove that if G is a countable nonamenable group then there exis...