arXiv subjects
Publications and source records attributed to Juris Steprans.
It will be shown to be consistent that there are at least two non-isomorphic selective ultrafilters, but no stable ordered-union ultrafilters. This answers a question of Blass from his 1987 paper which introduced the concept of a stable ordered-union ultrafilter.
In this series of papers we advance Ramsey theory of colorings over partitions. In this part, we concentrate on anti-Ramsey relations, or, as they are better known, strong colorings, and in particular solve two problems from [CKS21]. It is shown that for every infinite cardinal $\lambda$, a strong coloring on $\lambda^+$ by $\lambda$ colors over a partition can be stretched to one with $\lambda^{+}$ colors over the same partition. Also, a sufficient condition is given for when a strong coloring witnessing $Pr_1(\ldots)$ over a partition may be improved to witness $Pr_0(\ldots)$. Since the classical theory corresponds to the special case of a partition with just one cell, the two results generalize pump-up theorems due to Eisworth and Shelah, respectively.
The almost disjointness numbers associated to the quotients determined by the transfinite products of the ideal of finite sets are investigated. A $\mathrm{ZFC}$ lower bound involving the minimum of the classical almost disjointness and splitting numbers is proved for these characteristics. En route, it is shown that the splitting numbers associated to these quotients are all equal to the classical splitting number. Finally, it is proved to be consistent that the almost disjointness numbers associated to these quotients are all equal to the second uncountable cardinal while the bounding number is the first uncountable cardinal. Several open problems are considered.
The strongest type of coloring of pairs of countable ordinals, gotten by Todorcevic from a strongly Luzin set, is shown to be equivalent to the existence of a nonmeager set of reals of size $\aleph_1$. In the other direction, it is shown that the existence of both a strongly Luzin set and a coherent Souslin tree is compatible with the existence of a countable partition of pairs of countable ordinals such that no coloring is strong over it. This clarifies the interaction between a gallery of coloring assertions going back to Luzin and Sierpinski a hundred years ago.
In this series of papers, we advance Ramsey theory of colorings over partitions. In this part, a correspondence between anti-Ramsey properties of partitions and chain conditions of the natural forcing notions that homogenize colorings over them is uncovered. At the level of the first uncountable cardinal this gives rise to a duality theorem under Martin's Axiom: a function $p:[\omega_1]^2\rightarrow\omega$ witnesses a weak negative Ramsey relation when $p$ plays the role of a coloring if and only if a positive Ramsey relation holds over $p$ when $p$ plays the role of a partition. The consistency of positive Ramsey relations over partitions does not stop at the first uncountable cardinal: it is established that at any prescribed uncountable cardinal these relations follow from forcing axioms without large cardinal strength. This result solves in particular two problems from [CKS21].
A strong coloring on a cardinal $\kappa$ is a function $f:[\kappa]^2\to \kappa$ such that for every $A\subseteq \kappa$ of full size $\kappa$, every color $\gamma<\kappa$ is attained by $f\upharpoonright[A]^2$. The symbol $\kappa\nrightarrow [\kappa]^2_\kappa$ asserts the existence of a strong coloring on $\kappa$. We introduce the symbol $\kappa\nrightarrow_p[\kappa]^2_\kappa$ which asserts the existence of a coloring $f:[\kappa]^2\to \kappa$ which is strong over a partition $p:[\kappa]^2\to\theta$. A coloring $f$ is strong over $p$ if for every $A\in [\kappa]^\kappa$ there is $i<\theta$ so that every color $\gamma<\kappa$ is attained by $f\upharpoonright ([A]^2\cap p^{-1}(i))$. We prove that whenever $\kappa\nrightarrow[\kappa]^2_\kappa$ holds, also $\kappa\nrightarrow_p[\kappa]^2_\kappa$ holds for an arbitrary finite partition $p$. Similarly, arbitrary finite $p$-s can be added to stronger symbols which hold in any model of ZFC. If $\kappa^\theta=\kappa$, then $\kappa\nrightarrow_p[\kappa]^2_\kappa$ and stronger symbols, like $\mathrm{Pr}_1(\kappa,\kappa,\kappa,\chi)$ or $\mathrm{Pr}_0(\kappa,\kappa,\kappa,\aleph_0)$, hold also for an arbitrary partition $p$ to $\theta$ parts.
A function of two variables F(x,y)is universal iff for every other function G(x,y) there exists functions h(x) and k(y) with G(x,y) = F(h(x),k(y)) Sierpinski showed that assuming the continuum hypothesis there exists a Borel function F(x,y) which is universal. Assuming Martin's Axiom there is a universal function of Baire class 2. A universal function cannot be of Baire class 1. We show that it is consistent that for each countable ordinal alpha>2 there is a universal function of class alpha but none of smaller class. We show that it is consistent with ZFC that there is no universal function (Borel or not) on the reals, and we show that it is consistent that there is a universal function but no Borel universal function. We also prove some results concerning higher arity universal functions. For example, the existence of an F such that for every G there are unary h,k,j such that G(x,y,z) = F(h(x),k(y),j(z)) is equivalent to the existence of a 2-ary universal F. However the existence of an F such that for every G there are h,k,j such that G(x,y,z) = F(h(x,y),k(x,z),j(y,z)) follows from a 2-ary universal F but is strictly weaker. Results obtained Mar-June 2009, Nov 2010. Last revised April 2012 LaTex2e: 28 pages Latest version at: www.math.wisc.edu/~miller
Answering a question of the first author stated in [math.LO/0507519, 0.2] we show that limits of CS iterations of n$-Silver forcing notion have the n-localization property.
For a Polish group G let cov_G be the minimal number of translates of a fixed closed nowhere dense subset of G required to cover G. For many locally compact G this cardinal is known to be consistently larger than cov(meager) which is the smallest cardinality of a covering of the real line by meagre sets. It is shown that for several non-locally compact groups cov_G=cov(meager). For example the equality holds for the group of permutations of the integers, the additive group of a separable Banach space with an unconditional basis and the group of homeomorphisms of various compact spaces. Most recent version at: www.math.wisc.edu/~miller