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Consider the following statement (called $\Delta^d(\kappa^+, \lambda)$ where $d\in \omega$ and $\kappa<\lambda$ are cardinals): For every $A': [\lambda]^d\to [\lambda]^{\leq \kappa}$, there exist $E\i …

We can't find such pair of bad order types, aka there indeed is some Erdös-Rado phenomenon in the polarized partitions with respect to linear orderings. Indeed given $\gamma$ number of colors and $\th …

The answer is yes. In fact, more general statements are true. See https://arxiv.org/abs/1704.06827 for more detail (in particular Theorem 3.1).

In the countable context where we are given a perfect subtree $T$ of $2^{<\omega}$ and a sequence of colorings $f_i: T\to 2, i\in \omega$, it is possible to obtain a perfect subtree $T'\subset T$ and …

I believe the claim is wrong: If the claim is right I claim I can show $\alpha\to (\alpha)^2_2$ which is obviously wrong for countable ordinal $\alpha\geq \omega+2$. Given a coloring $f: [\alpha]^ …

The Halpern–Läuchli theorem theorem of dimension $d\in \omega+1$ is the following strong Ramsey theoretic statement: Given $k\in \omega$ and $d$ perfect finitely branching subtrees $T_i, i<d$ of …

It is known (a theorem of Komjáth and Hajnal) that it is consistent (by adding a Cohen real to a universe where there is no Suslin line) that there exists an order type $\theta$ such that for any othe …

Is there any example of a ($\omega_1$-)Aronszajn tree $T$ that is non-special and there exists a stationary subset $S\subset \omega_1$ such that $S$ is not stationary with respect to $T$? A tree bei …

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 $|f''[X]^{<\omega}|\leq \aleph_0$. Martin's Ma …

It is a theorem of Baumgartner and Laver that iterating Sacks forcings of weakly compact length gives rise to the tree property at $\omega_2$. Natural questions (at least for me) are: do we get strong …

The answer to the supercompact case is yes. More specifically, in the forcing extension obtained by iterating Sacks forcing of supercompact length, the super tree property at $\omega_2$ holds. This fo …

I believe you can use forcing to cook up many examples: Let $P=Add(\omega, 1)$. This is the forcing that adds a subset of $\mathbb{N}$. Let $\dot {U}$ be a $P$-name of a non-principal ultrafilter. We …

Aside from the well-known characterization of weakly compact cardinals in terms of the usual partition calculus, I've been wondering if there are other characterizations that are variants of the typic …

Regarding your comment before Question 3, you mean all such trees with size $<\mathfrak{m}$ are special. In particular, it doesn't say anything about trees of size $\geq 2^{\omega}$. Indeed, $T(\mathb …

Rado's conjecture holds in Mitchell's model (of course, start with a strongly compact instead of a weakly compact) granted the following: If $T$ if a non-special tree of height $\omega_1$, then $T$ re …