# 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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### Homogeneous Suslin tree in $L$ (Exercise 15.27 of Jech's Set Theory)

Exercise 15.27 of Jech's Set Theory says: If $V = L$ then there exists a homogeneous Suslin tree. Recall that a Suslin tree $T$ is a tree of height $\omega_1$ such that every chain and antichain is ...
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### A CSP on bit vector operations

I've got a CSP which is based on constraining bit vector variables. It is explained below through an example, followed by the full definition. So, what I'm concerned about is if you have some idea if ...
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### Finding where in Ramsey's theorem one uses the Axiom of choice

Ramsey's Theorem for infinite graphs requires some choice but when looking at the proof it is not evident how choice is exactly used. Sketch of the proof: Given $c:[\omega]^2\rightarrow 2$ a ...
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### Help with an exercise from Kunen's book

I've been having a hard time deciphering Kunen's suggestion in the following exercise: Overall, I'd like to get a little more direction in solving this exercise. Specifically, I would like some help ...
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### A probability question on infinite sets

Suppose all members of a countably infinite set of people each pick an integer "at random" (I want to avoid uniform distributions for the obvious reasons). (I hope this setting makes enough ...
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### On the distributive number $\mathfrak h$

The distributive number $\mathfrak h$ is defined as the least cardinal $\kappa$ such that there exists a family of $\kappa$ open dense subsets in the preordered set $([\omega]^\omega,\subset^*)$ whose ...
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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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### Partitioning an infinite set into fixed number of sets

Suppose we have a set of size $\kappa$, and want to partition it into $\mu$ sets, where $\kappa$ is an infinite cardinal, and $1<\mu\leq\kappa$. I am aware that it can be done in $2^\kappa$ ways (...
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### the generalized Delta System Lemma

I'm trying to understand the proof of the generalized Delta System Lemma in Kunen's Set Theory, the 2013 edition. This stationary set $T$ we've constructed using Fodor's Lemma (Pressing Down Lemma) ...
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### Infinite rooms and doors (2)

Suppose we have a house (with finitely many rooms) in which every room has an even number of doors. Prove that the number of doors from the house to the outside world is also even. All I could figure ...
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Given any directed graph $G=(V,R)$ where $R\subseteq V\times V$ is an arbitrary binary relation, we have under the standard definition of distance in an unweighted digraph that $d_G:V\times V\to \... • 10.4k 2 votes 0 answers 65 views ### Verifying a proof (under AC) that any (possibly non-finite) graph$G$has a$\kappa$-coloring ($\kappa$a cardinal) if$\chi(G)\leq \kappa\leq |V(G)|$Assuming the axiom of choice it seems pretty intuitive that every simple graph$G$has a$\kappa$-coloring if$\chi(G)\leq \kappa\leq |V(G)|$, simply color$G$with$\chi(G)$colors and swap out$\...
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There are theories that prove the existence and non-existence of Souslin trees [exist if $V=L$, don't exist if $\mathsf{MA}(\aleph_1)$] and Kurepa trees [exist if $V=L$, don't exist by Lévy collapsing ...