Articles of infinitary combinatorics

Extending almost disjoint family preserving the positivity of a member

An almost disjoint family is an infinite collection $\mathcal A$ of infinite subsets of $\omega$ such that for all $A, B \in \mathcal A$, the intersection $A \cap B$ is finite. A mad family is a maximal almost disjoint family. The free ideal generated by $\mathcal A$ is the collection $\mathcal I(\mathcal A)=\{X \subset \omega: […]

Dushnik-Miller Proof

I am working through the Dushnik-Miller Theorem in Jech’s Set Theory and have a question to one part of the proof. For the one’s who don’t have the book at their fingertips here Theorem 9.7: For every infinite cardinal $\kappa$ holds $\kappa\to (\kappa, \omega)^2$. For the proof he choses $\{A,B\}$ to be the partition of […]

The tree property for non-weakly compact $\kappa$

In my previous question, Weakly-compact cardinals, I was asking about weakly-compact cardinals and equivalent definitions to the basic one, which is $\kappa \to (\kappa)^2_2$. One of which was that $\kappa$ is inaccessible and has the tree property (that is if any tree of cardinality $\kappa$ for which every level is of cardinality $<\kappa$ then it […]

Disjoint Refinement

Prove that for any countable family of infinite sets from $\mathbb{N}$, $A = \{A_n \colon n \in \mathbb{N}\}$, there is a disjoint refinement $B = \{B_n \colon n \in \mathbb{N}\}$ of infinite sets, that is for every $n$, $B_n$ is an infinite subset of $A_n$, and all the $B_n$’s are pairwise disjoint. Also prove the […]

König's Infinity Lemma and Aronszajn Trees

I am working through the notes of my Set Theory lecture. There my professor wrote: ‘Is there an uncountable $\kappa$ such that König’s Infinity Lemma holds for $\kappa$? There are models where $\aleph_2$ is such. But, if CH holds then there are $\aleph_2$-Aronszajn trees…’ What could he mean with the three dots? What would follow […]

Dominating strategically $\omega_1$ reals

For a given $\kappa > \omega$, define the game $d(\kappa)$ that runs for $\omega$ stages as follows: At stage $n$, player I chooses a sequence of elements of $\omega$, $g_n$ of length $\kappa$, and player II picks a natural number $f(n)$. Define $h_{\alpha} (n) = g_n (\alpha)$. Player I wins the game if exists $\alpha […]

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 […]