1. CMB Online first
2. CMB 2017 (vol 61 pp. 812)
 Medini, Andrea; van Mill, Jan; Zdomskyy, Lyubomyr S.

Infinite Powers and Cohen Reals
We give a consistent example of a zerodimensional separable
metrizable space $Z$ such that every homeomorphism of $Z^\omega$
acts like a permutation of the coordinates almost everywhere.
Furthermore, this permutation varies continuously. This shows
that a result of Dow and Pearl is sharp, and gives some insight
into an open problem of Terada. Our example $Z$ is simply the
set of $\omega_1$ Cohen reals, viewed as a subspace of $2^\omega$.
Keywords:infinite power, zerodimensional, firstcountable, homogeneous, Cohen real, hhomogeneous, rigid Categories:03E35, 54B10, 54G20 

3. CMB 2014 (vol 57 pp. 579)
 Larson, Paul; Tall, Franklin D.

On the Hereditary Paracompactness of Locally Compact, Hereditarily Normal Spaces
We establish that if it is consistent that there is a
supercompact cardinal, then it is consistent that every locally
compact, hereditarily normal space which does not include a perfect
preimage of $\omega_1$ is hereditarily paracompact.
Keywords:locally compact, hereditarily normal, paracompact, Axiom R, PFA$^{++}$ Categories:54D35, 54D15, 54D20, 54D45, 03E65, 03E35 

4. CMB 2013 (vol 57 pp. 119)
5. CMB 2013 (vol 57 pp. 631)
 Sokić, Miodrag

Indicators, Chains, Antichains, Ramsey Property
We introduce two Ramsey classes of finite relational structures. The first
class contains finite structures of the form $(A,(I_{i})_{i=1}^{n},\leq
,(\preceq _{i})_{i=1}^{n})$ where $\leq $ is a total ordering on $A$ and $%
\preceq _{i}$ is a linear ordering on the set $\{a\in A:I_{i}(a)\}$. The
second class contains structures of the form $(A,\leq
,(I_{i})_{i=1}^{n},\preceq )$ where $(A,\leq )$ is a weak ordering and $%
\preceq $ is a linear ordering on $A$ such that $A$ is partitioned by $%
\{a\in A:I_{i}(a)\}$ into maximal chains in the partial ordering $\leq $ and
each $\{a\in A:I_{i}(a)\}$ is an interval with respect to $\preceq $.
Keywords:Ramsey property, linear orderings Categories:05C55, 03C15, 54H20 

6. CMB 2012 (vol 57 pp. 61)
 Geschke, Stefan

2dimensional Convexity Numbers and $P_4$free Graphs
For $S\subseteq\mathbb R^n$ a set
$C\subseteq S$ is an $m$clique if the convex hull of no $m$element subset of
$C$ is contained in $S$.
We show that there is essentially just one way to construct
a closed set $S\subseteq\mathbb R^2$ without an uncountable
$3$clique that is not the union of countably many convex sets.
In particular, all such sets have the same convexity number;
that is, they
require the same number of convex subsets to cover them.
The main result follows from an analysis of the convex structure of closed
sets in $\mathbb R^2$ without uncountable 3cliques in terms of
clopen, $P_4$free graphs on Polish spaces.
Keywords:convex cover, convexity number, continuous coloring, perfect graph, cograph Categories:52A10, 03E17, 03E75 

7. CMB 2012 (vol 56 pp. 709)
 Bartošová, Dana

Universal Minimal Flows of Groups of Automorphisms of Uncountable Structures
It is a wellknown fact, that the greatest ambit for
a topological group $G$ is the Samuel compactification of $G$ with
respect to the right uniformity on $G.$ We apply the original
description by Samuel from 1948 to give a simple computation of the
universal minimal flow for groups of automorphisms of uncountable
structures using FraÃ¯ssÃ© theory and Ramsey theory. This work
generalizes some of the known results about countable structures.
Keywords:universal minimal flows, ultrafilter flows, Ramsey theory Categories:37B05, 03E02, 05D10, 22F50, 54H20 

8. CMB 2011 (vol 56 pp. 564)
 Herzog, Ivo

Ziegler's Indecomposability Criterion
Ziegler's Indecomposability Criterion is used to prove that a totally
transcendental, i.e., $\Sigma$pure injective, indecomposable left
module over a left noetherian ring is a directed union of finitely
generated indecomposable modules. The same criterion is also used to
give a sufficient condition for a pure injective indecomposable module
${_R}U$ to have an indecomposable local dual $U_R^{\sharp}.$
Keywords:pure injective indecomposable module, local dual, generic module, amalgamation Categories:16G10, 03C60 

9. CMB 2011 (vol 56 pp. 317)
 Dorais, François G.

A Note on Conjectures of F. Galvin and R. Rado
In 1968, Galvin conjectured that an uncountable poset $P$ is the
union of countably many chains if and only if this is true for every
subposet $Q \subseteq P$ with size $\aleph_1$. In 1981, Rado
formulated a similar conjecture that an uncountable interval graph $G$ is countably
chromatic if and only if this is true for every induced subgraph $H
\subseteq G$ with size $\aleph_1$. TodorÄeviÄ has shown
that Rado's Conjecture is consistent relative to the existence of a
supercompact cardinal, while the consistency of Galvin's Conjecture
remains open. In this paper, we survey and collect a variety of
results related to these two conjectures. We also show that the
extension of Rado's conjecture to the class of all chordal graphs is
relatively consistent with the existence of a supercompact cardinal.
Keywords:Galvin conjecture, Rado conjecture, perfect graph, comparability graph, chordal graph, cliquecover number, chromatic number Categories:03E05, 03E35, 03E55 

10. CMB 2011 (vol 56 pp. 203)
 Tall, Franklin D.

Productively LindelÃ¶f Spaces May All Be $D$
We give easy proofs that (a) the Continuum Hypothesis implies that if
the product of $X$ with every LindelÃ¶f space is LindelÃ¶f, then $X$ is
a $D$space, and (b) Borel's Conjecture implies every Rothberger space
is Hurewicz.
Keywords:productively LindelÃ¶f, $D$space, projectively $\sigma$compact, Menger, Hurewicz Categories:54D20, 54B10, 54D55, 54A20, 03F50 

11. CMB 2011 (vol 55 pp. 378)
 Oman, Greg; Salminen, Adam

On Modules Whose Proper Homomorphic Images Are of Smaller Cardinality
Let $R$ be a commutative ring with identity, and let $M$ be a
unitary module over $R$. We call $M$ Hsmaller (HS for short) if and only if
$M$ is infinite and $M/N<M$ for every nonzero submodule $N$ of
$M$. After a brief introduction, we show that there exist nontrivial
examples of HS modules of arbitrarily large cardinality over
Noetherian and nonNoetherian domains. We then prove the following
result: suppose $M$ is faithful over $R$, $R$ is a domain (we will
show that we can restrict to this case without loss of generality),
and $K$ is the quotient field of $R$. If $M$ is HS over $R$, then
$R$ is HS as a module over itself, $R\subseteq M\subseteq K$, and
there exists a generating set $S$ for $M$ over $R$ with $S<R$.
We use this result to generalize a problem posed by Kaplansky and
conclude the paper by answering an open question on JÃ³nsson
modules.
Keywords:Noetherian ring, residually finite ring, cardinal number, continuum hypothesis, valuation ring, JÃ³nsson module Categories:13A99, 13C05, 13E05, 03E50 

12. CMB 2010 (vol 54 pp. 270)
 Dow, Alan

Sequential Order Under PFA
It is shown that it follows from PFA
that there is no
compact scattered space of height greater than $\omega$
in which the sequential order and the scattering heights coincide.
Keywords:sequential order, scattered spaces, PFA Categories:54D55, 03E05, 03E35, 54A20 

13. CMB 2010 (vol 53 pp. 286)
 Gorelic, Isaac

Orders of πBases
We extend the scope of B. Shapirovskii's results on the order of $\pi$bases in compact spaces and answer some questions of V. Tkachuk.
Keywords:Shapirovskii πbase, pointcountable πbase, free sequences, canonical form for ordinals Categories:54A25, 03E10, 03E75, 54A35 

14. CMB 2009 (vol 53 pp. 64)
 Dodos, Pandelis

On Antichains of Spreading Models of Banach Spaces
We show that for every separable Banach space $X$,
either $\mathrm{SP_w}(X)$ (the set of all spreading models
of $X$ generated by weaklynull sequences in $X$, modulo
equivalence) is countable, or $\mathrm{SP_w}(X)$ contains an
antichain of the size of the continuum. This answers
a question of S.~J. Dilworth, E. Odell, and B. Sari.
Categories:46B20, 03E15 

15. CMB 2009 (vol 52 pp. 303)
 Shelah, Saharon

A Comment on ``$\mathfrak{p} < \mathfrak{t}$''
Dealing with the cardinal invariants ${\mathfrak p}$ and
${\mathfrak t}$ of the continuum, we prove that
${\mathfrak m}={\mathfrak p} = \aleph_2\ \Rightarrow\ {\mathfrak t} =\aleph_2$.
In other words, if ${\bf MA}_{\aleph_1}$ (or a weak version of
this) holds, then (of course $\aleph_2\le {\mathfrak p}\le
{\mathfrak t}$ and) ${\mathfrak p}=\aleph_2\ \Rightarrow\
{\mathfrak p}={\mathfrak t}$. The proof is based on a criterion
for ${\mathfrak p}<{\mathfrak t}$.
Categories:03E17, 03E05, 03E50 

16. CMB 2009 (vol 52 pp. 127)
17. CMB 2008 (vol 51 pp. 579)
 Matet, Pierre

Guessing with Mutually Stationary Sets
We use the mutually stationary sets of Foreman and Magidor
as a tool to establish the validity of the twocardinal version
of the diamond principle in some special cases.
Keywords:$P_\kappa(\lambda)$, diamond principle Category:03E05 

18. CMB 2008 (vol 51 pp. 593)
19. CMB 2007 (vol 50 pp. 519)
 Henson, C. Ward; Raynaud, Yves; Rizzo, Andrew

On Axiomatizability of NonCommutative $L_p$Spaces
It is shown that Schatten $p$classes
of operators between Hilbert spaces of different (infinite)
dimensions have ultrapowers which are (completely) isometric to
noncommutative $L_p$spaces. On the other hand, these Schatten
classes are not themselves isomorphic to noncommutative $L_p$
spaces. As a consequence, the class of noncommutative $L_p$spaces
is not axiomatizable in the firstorder language developed by
Henson and Iovino for normed space structures, neither in the
signature of Banach spaces, nor in that of operator spaces. Other
examples of the same phenomenon are presented that belong to the
class of corners of noncommutative $L_p$spaces. For $p=1$ this
last class, which is the same as the class of preduals of ternary
rings of operators, is itself axiomatizable in the signature of
operator spaces.
Categories:46L52, 03C65, 46B20, 46L07, 46M07 

20. CMB 2002 (vol 45 pp. 71)
21. CMB 2000 (vol 43 pp. 397)
22. CMB 1999 (vol 42 pp. 13)
 Brendle, Jörg

Dow's Principle and $Q$Sets
A $Q$set is a set of reals every subset of which is a relative
$G_\delta$. We investigate the combinatorics of $Q$sets and
discuss a question of Miller and Zhou on the size $\qq$ of the smallest
set of reals which is not a $Q$set. We show in particular that various
natural lower bounds for $\qq$ are consistently strictly smaller than
$\qq$.
Keywords:$Q$set, cardinal invariants of the continuum, pseudointersection number, $\MA$($\sigma$centered), Dow's principle, almost disjoint family, almost disjointness principle, iterated forcing Categories:03E05, 03E35, 54A35 
