Some partition properties for measurable colourings of omega-one^2

Categoricity of theories in L_{kappa^* omega}, when kappa^* is a measurable cardinal. Part II

We continue the work of [KlSh:362] and prove that for lambda successor, a lambda-categorical theory T in L_{kappa^*, omega} is mu-categorical for every mu, mu <= lambda which is above the (2^{LS(T)})^+-beth cardinal.

局部有限 Borel 图的不友好划分

Unfriendly partitions of locally finite Borel graphs

本文否证了局部有限 Borel 图存在 Borel 不友好划分的猜想，构造无界度反例，并证明最大度至多四时存在不友好着色。

Global Square and Mutual Stationarity at the Aleph_n

We show using a proof of the Global Square property in Core Models below a measurable of Mitchell order o(kappa)=kappa^++ (a result originally due to Jensen & Zeman) that Foreman and Magidor's Mutual Stationarity property MS(Aleph_n (1<n<omega), Cof(omega_1)) implies the existence of inner models with measurables of high Mitchell order. This MS property states that any sequence of independently chosen stationary subsets S_n of the Aleph_n (of fixed cofinality omega_1) is mutually stationary below aleph_omega.

Cardinal characteristics at aleph omega

We prove the consistency of the statement $\mathfrak{u}_{\aleph_ω}<2^{\aleph_ω}$. We show that the consistency strength of this statement is exactly a measurable cardinal $μ$ so that $o(μ)=μ^{++}$.

Paradoxical decompositions and finitary rules

We colour every point x of a probability space X according to the colours of a finite list x_1, ...., x_k of points such that each of the x_i, as a function of x, is a measure preserving transformation. We ask two questions about a colouring rule (1) does there exist a finitely additive extension of the probability measure for which the x_i remain measure preserving and also a colouring obeying the rule almost everywhere that is measurable with respect to this extension?, and (2) does there exist any colouring obeying the rule almost everywhere? if the answer to the first question is no and to…

Set theory without choice: not everything on cofinality is possible

We prove (ZF+DC) e.g. : if mu =|H(mu)| then mu^+ is regular non measurable. This is in contrast with the results for mu = aleph_{omega} on measurability see Apter Magidor [ApMg]