Expand all Collapse all | Results 1 - 20 of 20 |
1. CMB Online first
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
pre-image of $\omega_1$ is hereditarily paracompact.
Keywords:locally compact, hereditarily normal, paracompact, Axiom R, PFA$^{++}$ Categories:54D35, 54D15, 54D20, 54D45, 03E65, 03E35 |
2. CMB 2013 (vol 57 pp. 119)
Splitting Families and Complete Separability We answer a question from Raghavan and SteprÄns
by showing that $\mathfrak{s} = {\mathfrak{s}}_{\omega, \omega}$. Then we use this to construct a completely separable maximal almost disjoint family under $\mathfrak{s} \leq \mathfrak{a}$, partially answering a question of Shelah.
Keywords:maximal almost disjoint family, cardinal invariants Categories:03E05, 03E17, 03E65 |
3. CMB Online first
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 |
4. CMB 2012 (vol 57 pp. 61)
2-dimensional 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 3-cliques 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 |
5. CMB 2012 (vol 56 pp. 709)
Universal Minimal Flows of Groups of Automorphisms of Uncountable Structures It is a well-known 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 |
6. CMB 2011 (vol 56 pp. 564)
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 |
7. CMB 2011 (vol 56 pp. 317)
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, clique-cover number, chromatic number Categories:03E05, 03E35, 03E55 |
8. CMB 2011 (vol 56 pp. 203)
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 |
9. CMB 2011 (vol 55 pp. 378)
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 $M$ H-smaller (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 non-Noetherian 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 |
10. CMB 2010 (vol 54 pp. 270)
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 |
11. CMB 2010 (vol 53 pp. 286)
Orders of π-Bases 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 π-base, point-countable π-base, free sequences, canonical form for ordinals Categories:54A25, 03E10, 03E75, 54A35 |
12. CMB 2009 (vol 53 pp. 64)
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 weakly-null 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 |
13. CMB 2009 (vol 52 pp. 303)
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 |
14. CMB 2009 (vol 52 pp. 127)
The Erd\H{o}s--Rado Arrow for Singular Cardinals We prove in ZFC that if $\cf(\lambda)>\aleph_0$ and
$2^{\cf (\lambda)}<\lambda$, then $\lambda \rightarrow
(\lambda,\omega+1)^2$.
Keywords:set theory, partition calculus Category:03E20 |
15. CMB 2008 (vol 51 pp. 579)
Guessing with Mutually Stationary Sets We use the mutually stationary sets of Foreman and Magidor
as a tool to establish the validity of the two-cardinal version
of the diamond principle in some special cases.
Keywords:$P_\kappa(\lambda)$, diamond principle Category:03E05 |
16. CMB 2008 (vol 51 pp. 593)
Chasing Silver We show
that limits of CS iterations of the $n$-Silver forcing notion have the
$n$-localization property.
Keywords:$n$-localization property, the Silver forcing, CS iterations Categories:03E40, 03E35 |
17. CMB 2007 (vol 50 pp. 519)
On Axiomatizability of Non-Commutative $L_p$-Spaces It is shown that Schatten $p$-classes
of operators between Hilbert spaces of different (infinite)
dimensions have ultrapowers which are (completely) isometric to
non-commutative $L_p$-spaces. On the other hand, these Schatten
classes are not themselves isomorphic to non-commutative $L_p$
spaces. As a consequence, the class of non-commutative $L_p$-spaces
is not axiomatizable in the first-order 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 non-commutative $L_p$-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 |
18. CMB 2002 (vol 45 pp. 71)
Images of Additive Polynomials in $\FF_q ((t))$ Have the Optimal Approximation Property We show that the set of values of an additive polynomial in several
variables with arguments in a formal Laurent series field over a
finite field has the optimal approximation property: every element in
the field has a (not necessarily unique) closest approximation in this
set of values. The approximation is with respect to the canonical
valuation on the field. This property is elementary in the language
of valued rings.
Categories:12J10, 12L12, 03C60 |
19. CMB 2000 (vol 43 pp. 397)
Tournaments and Orders with the Pigeonhole Property A binary structure $S$ has the pigeonhole property ($\mathcal{P}$) if
every finite partition of $S$ induces a block isomorphic to $S$. We
classify all countable tournaments with ($\mathcal{P}$); the class of
orders with ($\mathcal{P}$) is completely classified.
Keywords:pigeonhole property, tournament, order Categories:05C20, 03C15 |
20. CMB 1999 (vol 42 pp. 13)
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 |