# Questions tagged [infinitary-combinatorics]

For topics of a combinatorial character in set theory. Topics belonging to "combinatorial set theory" may be tagged this way. These include: Partition calculus, diamond principles, square principles, combinatorial properties of infinite graphs or partial orders, etc.

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### Is Fodor's Lemma necessary for the $\omega_1$ train station puzzle?

A common homework problem with Fodor's lemma is to imagine a train network with stations $\langle S_\alpha\mid \alpha<\omega_1+1\rangle$. The train starts at $S_0$ and stops at each station. At ...
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### Zorn's Lemma and injective choice functions

Let $X\neq\emptyset$ be a set and let ${\cal E}\subseteq {\cal P}(X)\setminus\{\emptyset\}$ be a collection of non-empty subset. We say that a map $f: {\cal E}\to X$ is an injective choice function if ...
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### A lemma in the proof of $\kappa$-chain condition preservation with iterated forcing

I do have a question about a fragment of a proof of a theorem in iterated forcing. It is the one that Jech invokes in the following form as theorem 16.30 in the 'Seth Theory' book and as Part II, Thm ...
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### If $\kappa$ is ineffable, then there is no $\kappa$-Kurepa tree

So this is an exercise($4.25$) from Ralf Schindler's book, and I have some trouble with it. This is the statement: (Jensen-Kunen) Show that if $\kappa$ is ineffable, then there is no $\kappa$-...
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### Showing that the least cardinal $\kappa$ such that a specific poset has no strong antichain of size $\kappa$ is regular

Given a poset $P$, let $\operatorname{cc}(P)$ be the least cardinal $\kappa$ such that $P$ has no strong antichains of size $\kappa$ — i.e. for any set of size $\kappa$, there exist two elements ...
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### The splitting number is uncountable

How can we show that any splitting family is uncountable? A family ${\cal F} \subset 2^\omega$ is called a splitting family if for every infinite $X \subset \omega$ there exists $A \in F$ such that ...
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### Understanding a proof about almost disjoint families

I am having trouble on understanding Michael Hrusak's proof on "Recent Progress in General Topology III", page 605. The theorem states the following: Given an AD family $\mathcal A$ and a ...
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### 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 ...
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### Kunen's proof of $\mathfrak{b}\le \mathfrak{a}$

I'm reading Kunen's Set Theory, and I'm stuck on his proof that least size of a dominating family of $\omega^\omega$, denoted by $\mathfrak{b}$, is less than or equal to the least size of a maximal ...
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### Recovering (reconstructing) all possible walks from a finite subset of walks over a finite graph

Compute a finite set of walks for a given finite graph $g$ such that the (possibly infinitely large) set of all (finitely long) walks over the same graph can be recovered (reconstructed).  Background ...
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### Do well pruned $\omega_1$ Aronszajn trees have any branches?

If a tree is well pruned then any node will have a node above it at any level of the tree above the nodes level. Doesn't this mean that no branch can have a countable height, because if you assume a ...
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### Stochastics in infinity

I am a German student who will finish his A-levels in around half a year, and for quite a long time I have this question on mind that I'm unable to find an answer to. You have a value y=100 Now, you ...
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### Free infinitary algebras

Let $\Sigma$ be a set which consists of function symbols with allowed infinite arities (which can be arbitrary sets). A $\Sigma$-algebra is a set $X$ equipped with maps $\omega_* : X^d \to X$ for each ...
### $\Box_{\kappa}$ principle, unknown derivation of its property
In Set Theory: The Third Millennium Edition, revised and expanded, page $443$, Jech states the $\Box_{\kappa}$ principle as follows: $(\Box_\kappa)$ There exists a sequence \$\langle C_\alpha:\alpha\...