# 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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### Find an equivalence relation over all of $\mathbb{Z}$ which has infinitely many equivalence classes with infinitely many elements in each

I want to find an equivalence relation defined on all integers (that is, all of $\mathbb{Z}$) where The equivalence relation partitions $\mathbb{Z}$ into infinitely many equivalence classes; and ...
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### Solving systems of simultaneous equations with permutations as variables: algebra meets combinatorics and order statistics

We may treat a given permutation as a vector variable, so that permutation s$_n$ is a variable that takes values among all possible number sequences of length $n$, which have elements in the rank ...
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### Forcing to add a $\square_\lambda$-sequence is $\mathrm{<}\lambda^+$-strategically closed

For an uncountable cardinal $\lambda,$ a $\square_\lambda$-sequence is a sequence $(C_\alpha: \alpha\in \lim(\lambda^+))$ such that Each $C_\alpha$ is a club in $\alpha$ with order type $\le \lambda.$...
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### Number of infinite trees on certain cardinal

Let $\kappa$ be some infinite cardinal. I usually use the notation $\langle T,<_T\rangle$ to refer to a tree. My question is: what is the number of non-isomorphic trees with $|T|=\kappa$? There is ...
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### Cardinality of tree of height $\kappa$, $\kappa$ measurable

The following is Theorem $2.8$ in Chapter 13 of Hrbacek and Jech's "Introduction to Set Theory" ($3^{\text{rd}}$ edition): Theorem $\bf{2.8}$ Let $\kappa$ be a measurable cardinal. If $T$ ...
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### Reference request: infinitary Ramsey theory

I was reading about the (various) infinite versions of Ramsey's theorem, and stumbled across a text containing a proof of the Bolzano-Weierstrass Theorem using it: Unfortunately, I forgot where I ...
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### Maximal almost disjoint family not countable

I recently stumbled upon the concept of almost disjoint families and I read that maximal almost disjointness requires at least an uncountable cardinality of the family. I believe it is not so clear to ...
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### Continuous chain of $\kappa^+$ isomorphic linear orders on $\kappa\ge\aleph_1$

For $\kappa\ge\aleph_0$ an infinite cardinal, and for $\kappa<\alpha\le\kappa^+$ an ordinal, we ask the existence of a chain $\left\langle(I_i,{<}):i<\alpha\right\rangle$ of linear orders ...
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### Club set in a finite support iteratiot of c.c.c forcing notions.

I'm reading the Baumgartner and Dordal paper "Adjoining dominating functions" and I have a problem with the Theorem 4.1 proof. My problem is the following: Suppose that $P$ is a c.c.c-...
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### Why is MA not provable from ZFC?

All texts I have seen about Martin's Axiom briefly mention that it is independent of ZFC, that it is implied by CH and that it is relatively consistent with ZFC+(not CH). I have seen proofs of the ...
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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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### Unexpected result, does $\Big\lfloor\frac{n-1}{2}\Big\rfloor=\sum_{i=1}^\infty\bigg\lfloor\frac{n+2^i-1}{2^{i+1}}\bigg\rfloor$

While trying to prove something else, I arrived at the result that for $n\in\Bbb{Z}^+$ \Big\lfloor\frac{n-1}{2}\Big\rfloor=\Big\lfloor\frac{n+1}{4}\Big\rfloor+\Big\lfloor\frac{n+3}{8}\Big\rfloor+\...
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### Colorability of graphs with finite clique number

Let $G$ be an infinite graph such that the clique number $\omega(G)$ is finite. By the De Brujin - Erdos theorem, the chromatic number $\chi(G)$ is finite if and only there is a bound on the chromatic ...
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### Generalisation of infinite Ramsey's theorem to countably infinite or variable edge arities

The usual statement of the infinite Ramsey's Theorem, as appears e.g. in Wikipedia page on Ramsey's Theorem is (paraphrasing slightly): If $X$ is an infinite set, and $C$ is a finite set, and $n$ ...
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Assume that $\kappa \to (\lambda)^r_s$ holds. Prove that if $r'\le r$, $\kappa \to (\lambda)^{r'}_s$ I proved similar cases like when $s'\le s$ etc. but I have no idea about this case. Actually I can'...