# 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.

264 questions
Filter by
Sorted by
Tagged with
40 views

### 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 $T$ ...
• 117
37 views

### 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 ...
• 1,311
16 views

### 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 ...
• 25
304 views

### 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 ...
• 174
1 vote
56 views

• 128
217 views

• 2,592
1 vote
37 views

### 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-...
• 2,592
1 vote
138 views

### 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 ...
• 126
352 views

### 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 ...
• 1,466
194 views

### 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 ...
226 views

### 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 ...
• 135
80 views

### 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 $\kappa$-...
• 2,324
108 views

### 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+\...
41 views

### 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 ...
• 2,865
241 views

### 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 $n$ ...
• 180
1 vote
83 views

### Simple property of arrow notation

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'...
• 196
65 views

• 367
This question arises from a previous answer of mine, in which I claimed a fact I am not so sure about. Let $\alpha\in(1,2)$ be an irrational number (feel free to assume $\alpha=\sqrt{2}$ or $\alpha=\... • 347k 6 votes 1 answer 106 views ### On the distributive number$\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^*)$whose ... • 297 1 vote 2 answers 145 views ### How is$\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$The way he explained it, ... -1 votes 1 answer 28 views ### Partitioning an infinite set into fixed number of sets Suppose we have a set of size$\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 (... 3 votes 2 answers 422 views ### 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) ... • 927 1 vote 1 answer 51 views ### A Bijection Between$\kappa\times\aleph_{\alpha+\kappa}$and$\aleph_{\alpha+\kappa}$with Certain Order-Preservation Properties In Claim 5.5 of the book$ AlmostFreeModules$, second edition, by P. Eklof and A. Mekler (page 254), the authors consider regular cardinals$\kappa$and$\aleph_\alpha$, where$\kappa<\...
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 ...