By an ${\omega }_1$- tree we mean a tree of power ${\omega }_1$ and height ${\omega }_1$. Under CH and $2^{{\omega }_1}>{\omega }_2$ we call an ${\omega }_1$-tree a Jech-Kunen tree if it has κ-many branches for some κ strictly between ${\omega }_1$ and $2^{{\omega }_1}$. In this paper we prove that, assuming the existence of one inaccessible cardinal, (1) it is consistent with CH plus $2^{{\omega }_1}>{\omega }_2$ that there exist Kurepa trees and there are no Jech-Kunen trees, which answers a question of [Ji2], (2) it is consistent with CH plus $2^{{\omega }_1}={\omega }_4$ that there only exist Kurepa trees with ${\omega }_3$-many branches, which answers another...

We show that if $X$ is a ${\Sigma }_1^1$ separable metrizable space which is not $\sigma$-compact then $C_p^{*}\left(X\right)$, the space of bounded real-valued continuous functions on $X$ with the topology of pointwise convergence, is Borel-${\Pi }_1^1$-complete. Assuming projective determinacy we show that if $X$ is projective not $\sigma$-compact and $n$ is least such that $X$ is ${\Sigma }_n^1$ then $C_p\left(X\right)$, the space of real-valued continuous functions on $X$ with the topology of pointwise convergence, is Borel-${\Pi }_n^1$-complete. We also prove a simultaneous improvement of theorems of Christensen...

If G is a group then the abelian subgroup spectrum of G is defined to be the set of all κ such that there is a maximal abelian subgroup of G of size κ. The cardinal invariant A(G) is defined to be the least uncountable cardinal in the abelian subgroup spectrum of G. The value of A(G) is examined for various groups G which are quotients of certain permutation groups on the integers. An important special case, to which much of the paper is devoted, is the quotient of the full symmetric group by the...

We show that under the axiom $CP{A}_{cube}$ there is no uniformly completely Ramsey null set of size $2^{\omega }$. In particular, this holds in the iterated perfect set model. This answers a question of U. Darji.

We study a notion of potential isomorphism, where two structures are said to be potentially isomorphic if they are isomorphic in some generic extension that preserves stationary sets and does not add new sets of cardinality less than the cardinality of the models. We introduce the notion of weakly semi-proper trees, and note that there is a strong connection between the existence of potentially isomorphic models for a given complete theory and the existence of weakly semi-proper trees. ...

We consider prediction problems in which each of a countably infinite set of agents tries to guess his own hat color based on the colors of the hats worn by the agents he can see, where who can see whom is specified by a graph V on ω. Our interest is in the case in which 𝓤 is an ultrafilter on the set of agents, and we seek conditions on 𝓤 and V ensuring the existence of a strategy such that the set of agents guessing correctly is of 𝓤-measure one. A natural necessary condition is the absence...

It is well known that if a nontrivial ideal ℑ on κ is normal, its quotient Boolean algebra P(κ)/ℑ is ${\kappa }^{+}$-complete. It is also known that such completeness of the quotient does not characterize normality, since P(κ)/ℑ turns out to be ${\kappa }^{+}$-complete whenever ℑ is prenormal, i.e. whenever there exists a minimal ℑ-measurable function in ${}^{\kappa }\kappa$. Recently, it has been established by Zrotowski (see [Z1], [CWZ] and [Z2]) that for non-Mahlo κ, not only is the above condition sufficient but also necessary for P(κ)/ℑ...

It is a well-known fact that modules over a commutative ring in general cannot be classified, and it is also well-known that we have to impose severe restrictions on either the ring or on the class of modules to solve this problem. One of the restrictions on the modules comes from freeness assumptions which have been intensively studied in recent decades. Two interesting, distinct but typical examples are the papers by Blass [1] and Eklof [8], both jointly with Shelah. In the first case the authors...

Diverse classes of fuzzy relations such as reflexive, irreflexive, symmetric, asymmetric, antisymmetric, connected, and transitive fuzzy relations are studied. Moreover, intersections of basic relation classes such as tolerances, tournaments, equivalences, and orders are regarded and the problem of preservation of these properties by $n$-ary operations is considered. Namely, with the use of fuzzy relations $R_1,...,R_n$ and $n$-argument operation $F$ on the interval $[0,1]$, a new fuzzy relation $R_F=F(R_1,...,R_n)$ is created. Characterization...

Let X be a Borel subset of the Cantor set C of additive or multiplicative class α, and f: X → Y be a continuous function onto Y ⊂ C with compact preimages of points. If the image f(U) of every clopen set U is the intersection of an open and a closed set, then Y is a Borel set of the same class α. This result generalizes similar results for open and closed functions.

I isolate a simple condition that is equivalent to preservation of P-points in definable proper forcing.