1. 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
2. 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
3. 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
4. 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