All Questions
12 questions with no upvoted or accepted answers
12
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0
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213
views
Is $\kappa \rightarrow [\kappa]^2_3$ the same as $\kappa \rightarrow [\kappa]^2_2$ for inaccessible $\kappa$
The principle $\kappa \rightarrow [\kappa]^2_\alpha$ states that whenever we have a coloring $c:[\kappa]^2\rightarrow \alpha$ there is $H \subset \kappa$ of size $\kappa$ s.t. $|c"[H]^2|<\alpha$.
...
- 490
11
votes
0
answers
490
views
$\Sigma^2_1$ and the Continuum Hypothesis
This is a follow up to Will Brian's answer to this recent question. In particular, quoting Brian:
"In fact, Paul Larson has pointed out to me that the statement "$\phi$ and $\phi^{-1}$ are conjugate"...
- 7,125
11
votes
0
answers
272
views
Preservation of chain condition under strategically closed forcing
It is well-known that $\kappa$-closed forcing preserves $\kappa$-c.c. posets. The same argument works for $\kappa$-strategically closed forcing. Here is the definition:
A poset $\mathbb P$ is $\...
- 18.6k
10
votes
0
answers
283
views
Martin's Maximum implies stationary/club Chang's conjecture?
Chang's Conjecture (CC) states: for any $f: [\omega_2]^{<\omega} \to \omega_1$, there exists a set $X\subset \omega_2$ of order type $\omega_1$ such that $|f''[X]^{<\omega}|\leq \aleph_0$.
...
- 3,038
9
votes
0
answers
250
views
Distributivity of certain infinite products
Suppose we have a sequence of posets $\{\mathbb P_n : n\in\omega\}$ such that for each $n$, $\mathbb P_{n+1}$ is $|\mathbb P_n|^+$-distributive. Is $\prod_{n>0} \mathbb P_n$ necessarily $|\mathbb ...
- 18.6k
9
votes
0
answers
162
views
Strong Chains in $^{\omega_1}\omega_1$ of length $\omega_3$
In a previous question, I asked about the impact of strong chains in $^{\omega_1}\omega_1$ (e.g., sequences of functions $\langle f_\alpha:\alpha<\kappa\rangle$ in $^{\omega_1}\omega_1$ that are ...
- 7,125
9
votes
0
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388
views
On the role of $\diamondsuit$
The well-known axiom $\diamondsuit$ states that there is a sequence $\langle A_\alpha:\alpha<\omega_1\rangle$ (a $\diamondsuit$-sequence) of countable sets with the property that for any $A\...
- 7,125
8
votes
0
answers
206
views
ladder system uniformization at successors of singulars
Shelah proved (paper 667) that if GCH holds and $\lambda$ is singular, then for every stationary $S \subseteq \{ \alpha < \lambda^+ : \text{cf}(\alpha) = \text{cf}(\lambda) \}$, there is a ladder ...
- 18.6k
8
votes
0
answers
241
views
Topological applications of $\mathfrak{p}=\mathfrak{t}$
I'm working on the Malliaris-Shelah's recent result of $\mathfrak{p}=\mathfrak{t}$, but I'm more interested in what possible topological applications can derivate from this equality.
Searching in ...
- 181
8
votes
0
answers
191
views
Specializing fat trees
The discussion is about trees of height $\omega_1$ that are not necessarily thin, namely, no cardinality constraints on the size of each level. A classcial theorem of Baumgartner states that it is ...
- 1,006
6
votes
0
answers
146
views
Combinatorial characterizations of potentially countably chromatic graphs
Is there a combinatorial characterization of (uncountably chromatic) graphs that are "potentially" countably chromatic? By this I mean: $G=(V,E)$ is a graph such that there exists a cardinal ...
- 1,006
2
votes
0
answers
69
views
Diamonds on supercompact $\kappa$ after a $\kappa$-c.c. forcing
Let $\kappa$ be supercompact. Then the (supercompact) Laver diamond holds at $\kappa$: There is $f:\kappa\to V_\kappa$ such that for all $\lambda\geq \kappa$ and $x\in H(\lambda^+)$ there is $j:V\to M$...
- 51