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1. 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 |
2. 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 |
3. 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 |