# 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.

220 questions
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### How is $\sum_{i=0}^{\infty}\sum_{j=0}^{\infty}x^{i+j} = \sum_{n=0}^{\infty}\sum_{i=0}^{n}x^n$

My professor used this in class for a proof and I'm having trouble understanding it. $\sum_{i=0}^{\infty}\sum_{j=0}^{\infty}x^{i+j} = \sum_{n=0}^{\infty}\sum_{i=0}^{n}x^n$ The way he explained it, ...
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### An infinite basis for an ad hoc vector space

To prove a combinatorial identity I'm building an ad hoc vector space that I will define below. First, I define the set of Pseudo Power Series. A Pseudo Power Series (pps) $\alpha$ is an ...
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### Universal Classes of Regular Graphs

There is a small but well-known open problem in graph coloring called the Total Coloring Conjecture. I worked on this problem several years ago and am interested in returning to it, potentially. For ...
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### Prove using Koenig's lemma that a rayless connected graph $G$ with $\delta(G) = k$ has a finite subgraph $H$ with $\delta(H)=k$

As the question says: I want to prove using Koenig's lemma that a rayless connected Graph $G$ with minimum degree $\delta(G) = k$ has a finite subgraph $H$ with $\delta(H) = k$. So obviously this is ...
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### Special tree properties

Let $κ$ be an uncountable regular cardinal and $T$ be a tree of height $\kappa$. (1) A function $r : T → T$ is called regressive if $r(t) <_T t$ holds for every $t \in T$ \ {root($T$)}. (2) The ...
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### Can the Infinite Ramsey Theorem be extended even when coloring a graph's edges with an infinite number of colors?

In the language of partition calculus, I want to know if the following is true: Let $n$ be an integer, $n \geq 2$. Then $\aleph_n \rightarrow (\aleph_0)^2_{\aleph_0}$. I'm new to infinitary ...
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### 0,1-Tree, number of branches

How many branches has this tree: $\{s\in 2^{<\kappa}:|\alpha\in dom(s):s(\alpha)\neq 0|<\aleph_0\}$, and why? How does the tree look like, i.e. what is its shape? Thank you.
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### Is $Ord$ weakly compact?

$\DeclareMathOperator{\Ord}{Ord}$Let $c: \Ord^{} \rightarrow 2$ be a $2$-coloring of the class of pairs of ordinals. Is there a definable class-sized subclass $H$ of $\Ord$ which is $c$-...
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### Shrinking a family of sets preserving cardinality

I'm having trouble with a simple problem about infinitary combinatorics. Suppose $\kappa$ is an infinite cardinal, $\mathcal A=\{A_\alpha: \alpha<\kappa\}$ and for each $\alpha$, $|A_\alpha|=\kappa$...
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### Version of Fodor's Lemma

I know that pressing down functions exist, but it is not intuitively clear to me. Is there a way to prove the existence of such functions in this form: $$f: \omega_1 \to \omega_1$$ \exists \alpha \...
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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 ...
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### special Aronszajn trees

I have found two different definitions for Aronszajn special trees: somewhere they say they are Aronszajn trees where there exists a strictly increasing function from the tree to the rational numbers, ...
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### Ineffable cardinals and diamond principle.

A regular uncountable cardinal $\kappa$ is said to be ineffable iff for every $\{A_\alpha\}_{\alpha<\kappa}$ such that $A_\alpha\subset \alpha$ there exists a stationary set $S$ verifying that for ...
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### Kurepa trees and inaccessible cardinals in $L$

Given a regular uncountable cardinal $\kappa$ we say that a $\kappa-$tree is $\kappa-$Kurepa if it has at least $\kappa^+$ branches. If $\kappa=\omega_1$ we simply say that $T$ is Kurepa. In this ...
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### Prikry Hypothesis implies Kurepa Hypothesis

I want to show that $\textbf{HP}_\kappa$ implies $\textbf{HK}_\kappa$, Prikry hypothesis and Kurepa hypothesis in $\kappa$, respectively. Where \begin{equation} \textbf{HK}_\kappa: \text{ There's a ...
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