Articles of infinitary combinatorics

What do we call well-founded posets whose elements have a unique height?

What do we call those well-founded posets $P$ with the property that for every $x \in P$, all maximal chains in the lowerset generated by $x$ have the same length? Examples: The set of all finite subsets of a (possibly infinite) set. The set of all finite-dimensional vector subspaces of a (possibly infinite-dimensional) vector space. […]

A question from Kunen's book: chapter VII (H9), about diamond principle

Assume $(\mathbb{P}$ is c.c.c.$)^M$ and $\Diamond$ holds in $M[G]$. Show that $\Diamond$ holds in $M$. Hint: It is sufficient to verify $\Diamond^-$ in $M$. Should I try to create a $\Diamond$ sequence or $\Diamond^{-}$ sequence? Should I work with nice names?

Cardinality of the set of ultrafilters on an infinite Boolean algebra

Let $\mathfrak B$ be a Boolean algebra with an infinite power $\kappa$. My question is how many ultrafilters does it have? $\kappa$ or $2^\kappa$? Or even smaller?

Aronszajn Trees and König's Lemma

I’m looking at König’s Lemma and Aronszajn Trees. I’ve seen the following link: König's Infinity Lemma and Aronszajn Trees Long story short, I’m still left with two questions and I was wondering if anyone would either be able to point me in the right direction or explain why. I apologise if they’re silly questions, I’m […]

A special cofinal family in $(\omega^\omega,\le)$

Let me start by posting a question, which is essentially a combinatorial question. I will post below at least a short explanation what lead me to ask this question. Let us work with the set $\omega^\omega$ (i.e., all sequences of positive integers) with a pointwise ordering $$x\le y \Leftrightarrow (\forall n\in \omega) x_n\le y_n$$ Does […]

Bartoszyński's results on measure and category and their importance

I have seen this interesting paragraph on a talk page of the Wikipedia article about Polish mathematician Tomek Bartoszyński: Tomek’s paper “Additivity of measure implies additivity of category, Trans. Amer. Math. Soc., 281(1984), no. 1, 209–213″ as the first to show that there is (in ZFC) an asymmetry between measure and category. Before that essentially […]

How to show that every Suslin tree is Frechet-Urysohn

A topological space $X$, is Frechet-Urysohn, if, given $x \in \overline A \subset X$, there exists a sequence of points in $A$ which converges to $x$. I am trying to prove that every Suslin tree, is Frechet-Urysohn. (I have read in an article, that this claim is folklore). Any help? Thank you!

A “geometrical” representation for Ramsey's theorem

The [infinite] Ramsey theorem states that Let $n$ and $k$ be natural numbers. Every partition $\{X_1,\ldots ,X_k\}$ of $[\omega]^n$ into $k$ pieces has an infinite homogeneous set. Equivalently, for every $F\colon [\omega]^n\to \{1, . . . , k\}$ there exists an infinite $H \subseteq \omega$ such that $F$ is constant on $[H]^n$. Where $[X]^k := […]

Equivalences of $\diamondsuit$-principle.

I’m working on exercises from Kunen and I’m stuck. I must proof that the following are equivalent: There exists a sequence $\langle A_\alpha:\alpha<\omega_1\rangle$ such that $\forall\alpha(A_\alpha\subset\alpha)$ and for all $A \subset \omega_1$ the set $\{\alpha \in \omega_1:A\cap\alpha=A_\alpha\}$ is stationary. ($\diamondsuit$-principle) There exists a sequence $\langle A_\alpha:\alpha<\omega_1\rangle$ such that $\forall\alpha(A_\alpha\subset\alpha\times\alpha)$ and for all $A \subset \omega_1\times\omega_1$ […]