The infinite-combinatorics tag has no usage guidance.

**0**

votes

**1**answer

63 views

### On some examples of critical families

I'm reading the book on Injective choice functions by Holz, Podewski and Steffens, and I find it to be at the same time well written and quite difficult. It has almost no examples - and in quite a few ...

**9**

votes

**0**answers

282 views

### What ccc forcings add a Suslin tree?

In a comment to Miha's question in Forcing PFA with ccc forcing, I suggested that if such situation is even possible, it might be achieved by screwing with PFA by some ccc forcing (e.g. adding a Cohen ...

**-1**

votes

**1**answer

40 views

### Size of smallest set in critical covering of $\omega$

A covering of a non-empty set $X$ is a collection ${\cal U} \subseteq ({\cal P}(X)\setminus\{\emptyset\})$ such that $\bigcup {\cal U} = X$. If ${\cal U}$ is a covering of $X$ then a function $f:{\cal ...

**0**

votes

**1**answer

84 views

### Critical coverings of $\omega$

A covering of a non-empty set $X$ is a collection ${\cal U} \subseteq ({\cal P}(X)\setminus\{\emptyset\})$ such that $\bigcup {\cal U} = X$. If ${\cal U}$ is a covering of $X$ then a function $f:{\cal ...

**5**

votes

**1**answer

250 views

### A kind of anti-Ramsey result

In contrast to classic results for arithmetic progressions of arbitrary length in one set at least of any finite partition of IN, it is easy to construct a partition in two sets of integers A and B ...

**11**

votes

**2**answers

860 views

### On Hamkins' answer to a problem by Michael Hardy

Based on a post by Michael Hardy and Hamkins' answer to it Andreas Blass, Will Brian, Joel Hamkins, Michael Hardy and Paul Larson introduced a new cardinal characteristic of the continuum $\mathfrak{...

**5**

votes

**1**answer

160 views

### almost disjoint ladder system on $\omega_2$

Suppose $\langle s_\alpha : \alpha \in \omega_2 \cap \mathrm{cof}(\omega_1) \rangle$ is a sequence such that each $s_\alpha$ is an increasing cofinal map from $\omega_1$ to $\alpha$. Is it possible ...

**2**

votes

**1**answer

174 views

### Injective subset function

Let $X$ be a non-empty set and let $F: X \to {\cal P}(X)$ be a function with the following property:
for $A \subseteq X$ we have $|A| \leq |\bigcup F(A)|$.
Does this imply that there is an ...

**10**

votes

**2**answers

402 views

### Constructing an $\omega_1$-sequence of functions that almost extend all previous functions

I want to construct a sequence of functions $$f_\alpha: \alpha \rightarrow \omega,\ \alpha < \omega_1$$
such that for all $\alpha < \omega_1$ the following holds:
$f_\alpha$ is injective.
$(\...

**9**

votes

**1**answer

328 views

### Can there be a tree of height $\omega_2$ having all levels countable, with no cofinal branch?

For many years I had the idea that if a well-founded tree is both very tall and very narrow, then it must have a cofinal branch. For example, it is a fun exercise to show that any $\omega_1$-tree ...

**25**

votes

**1**answer

1k views

### How hard is it to destroy a diamond? (with a real)

If we start with $V\models\lozenge$, it is not hard to force the failure of diamond. You can blow up the continuum, or destroy all the Suslin trees. You can blow up the continuum of $\aleph_1$, and ...

**2**

votes

**0**answers

66 views

### Edge-disjoint path-systems in infinite digraphs

Let $ D=(V,A) $ be a directed graph without backward-infinite paths and let $ \{ s_i \}_{i<\lambda},U \subset V $ where $
\lambda $ is some cardinal. Assume that for all $ u\in U $ there is a ...

**34**

votes

**1**answer

2k views

### Hilbert's Hotel

Hilbert's Hotel is a famous story about infinity attributed to David Hilbert (1862-1943).
Is it documented that Hilbert's Hotel is in fact due to Hilbert, and if yes, where?

**4**

votes

**1**answer

256 views

### A Question Regarding Weak Diamond

In Assaf Rinot's survey article "Jenson's diamond principle and its relatives", he proves the following fact:
Fact 2.5:For every stationary set S, $\Phi_{S}$...entails that no ladder system <$L_{\...

**6**

votes

**1**answer

387 views

### A “good scale” that is not really a scale

I don't know much about singular cardinal combinatorics, so I apologize in advance if I write something that is wrong or looks funny. First let me recall some basic definitions.
Let $\lambda$ be a ...

**24**

votes

**2**answers

1k views

### Does the symmetric group on an infinite set have a minimal generating set?

To clarify the terms in the question above:
The symmetric group Sym($\Omega$) on a set $\Omega$ consists of all bijections from $\Omega$ to $\Omega$ under composition of functions. A generating set $...

**7**

votes

**1**answer

384 views

### On Consistency of an Existence

Let $\omega \leq \kappa <2^{\omega}$ , $\omega \leq\lambda \leq \kappa$ and $D(\kappa, \lambda)$ be the statement:
For all $ \mathfrak{B} \subseteq \mathbf{P}(\omega)$ with $|\mathfrak{B}|=\kappa$...

**6**

votes

**1**answer

412 views

### regularity of ultrafilters

An ultrafilter $U$ is $(\mu,\kappa)$-regular if there is a sequence $\langle X_\alpha : \alpha < \kappa \rangle \subseteq U$ such that for all $y \in [\kappa]^\mu$, $\bigcap_{\alpha \in y} X_\alpha ...

**9**

votes

**1**answer

566 views

### Order homomorphism functions on $\omega_1$

Let $\omega_1$ be the first uncountable ordinal,
same as the set of all countable ordinals.
Let $F$ be the set of all functions
$f$ from $\omega_1$ minus singleton $0$ into $\omega_1$ that
are (a) ...

**6**

votes

**2**answers

924 views

### If $G \times G \cong H \times H$, then is $G \cong H$? [duplicate]

Let $G,H$ groups. Suppose that $G \times G \cong H \times H$. Then is necessarily $G \cong H$?
I know that if $G,H$ are finite groups then it is true (you show that the map $FinGrp \rightarrow \...

**7**

votes

**2**answers

357 views

### How to show that the chromatic number > aleph_0

Let $V =\{ f | \exists _{\alpha<\omega_1} ( f:\alpha \rightarrow \mathbb{N} \wedge f $ is $ 1-1) \}$. We define $E\subseteq [V]^2 $, such that $\forall_{f,g\in V } (<f,g>\in E \...

**7**

votes

**1**answer

293 views

### A special c.c.c forcing notion and adding minimal generic reals

This question is related to my question "Forcing with c.c.c forcing notions, Cohen reals and Random reals".
A natural way to answer Prikry's conjecture is to build a c.c.c. forcing notion which adds ...

**8**

votes

**1**answer

346 views

### Weak threads in $\square (\kappa, <\kappa)$ sequences

The following definition is well known ($\kappa$ is regular uncountable cardinal):
Definition: a sequence $\mathcal{C} = \langle \mathcal{C}_\alpha | \alpha < \kappa,\,\alpha \text{ limit ordinal}...

**4**

votes

**1**answer

243 views

### Adding large sets not containing countable ground model sets

The question is motivated by Toni's question "Approximation of infinite set in generic extension" (see Approximation of infinite set in generic extension).
Before I state the question, let me add ...

**5**

votes

**1**answer

159 views

### About non-stationary sets of $\omega_1$

Suppose $A$ is a non stationary set of $\omega_1$. Define by induction the following sequence of sets:\
$A_0 = A$
$A_{\alpha+1} = A_{\alpha}'$ [$X'$ is the subset of $X$, of all points the are ...

**6**

votes

**2**answers

164 views

### Separation of almost disjoint families by ground model almost disjoint families

Suppose that $V$ is a model of $\sf ZFC$, and for concreteness I should point that at this point I am interested in $V=L$ as a ground model.
Suppose that $V[c]$ is a Cohen extension of $V$ where $c$ ...

**11**

votes

**1**answer

468 views

### Are there insane families in $L$?

Let $A,B\subseteq\omega$. We write $A\subseteq^*B$ if $A\setminus B$ is finite, if additionally $B\setminus A$ is infinite then we write $A\subsetneq^*B$, otherwise we write $A=^*B$.
We say that a $\...

**17**

votes

**0**answers

850 views

### Hrushovski's Construction

Zilber expressed a conjecture for $\aleph_{1}$- categorical theories (In the 80s).
Zilber's Conjecture: The geometry of any $\aleph_{1}$- categorical structure is one of the following:
(a) Trivial (...

**6**

votes

**1**answer

690 views

### Is there a monster behind the trees?

First Fix the following notation:
$\forall \kappa\in Card~~~Tp(\kappa):="\kappa~has~tree~property"$
The large cardinals as "monsters of heaven" live everywhere in the land of ...

**3**

votes

**0**answers

76 views

### Hindman's theorem variant for noncommutative semigroups

The well set proof of Hindman's finite sums theorem applied to noncommutative semigroups yields a sequence of elements such that finite products ordered coherently with this sequence are in one set. ...

**6**

votes

**1**answer

279 views

### A problem about Ramsey Property

It is known that Ramsey property is a kind of generalizition of pigeon hole principle, and some kinds of Ramsey properties have lots of equivalent forms.
We often deal with the case $a\rightarrow (b)^...

**7**

votes

**1**answer

318 views

### Determinacy from $\omega_1\rightarrow(\omega_1)^{\omega_1}$

Assuming the Axiom of Determinacy (abbreviated AD), Martin showed how to derive a rather strong partition on $\omega_1$, namely that $\omega_1\rightarrow(\omega_1)^{\omega_1}$. In "Infinitary ...

**3**

votes

**1**answer

173 views

### Countable coloring of a plane

How does one prove existence decomposition of $R^2$ for countable many subsets $A(n)$
s
.t.$\forall$ $x,y$ $\epsilon$ $A(n)$ $|x-y|$ is nonrational?
I tried thinking of $R^2$ as infinite tree with $...

**9**

votes

**1**answer

532 views

### Does every strictly increasing, unbounded sequence of positive real numbers contain arbitrarily long, finite subsequences which are “sort of increasing” or “sort of decreasing” (as defined below)?

Is the following true?
If $(x_0, x_1, \dots)$ is a strictly increasing, unbounded sequence of positive real numbers, then there exist fixed $M,N \geq 1$ such that the sequence $(x_0, x_1, \dots)$ ...

**5**

votes

**2**answers

614 views

### Partition calculus question

For $m,n,k < \omega$, consider the equation
$X \to (\omega \times k)^{m}_{n}$
What is the smallest $X$ known to satisfy it?
Baumgartner-Hajnal theorem gives a satisfactory answer for $m=2$, but ...

**7**

votes

**2**answers

788 views

### Size of stationary sets

What can we say about the size of stationary subsets of $P_{\kappa}(\lambda)$ for infinite cardinals $\kappa, \lambda,$ especially when $\kappa=\aleph_1.$
Please give me some references, if there are ...

**24**

votes

**5**answers

1k views

### Game involving 'asking questions about a real'

Consider the following game, played by two players,
called Q and A, in a time frame t = 1, 2, ....
At every time point i, Q mentions some $Q_i \subset \mathbb{R}$,
after which A mentions $A_i$ such ...

**11**

votes

**0**answers

239 views

### Combinatorial Hilbert spaces

Any closed subspace $V\subset {\ell}^2(\omega)$ has associated to it a subset ${\cal S}_V$ of ${\cal P}(\omega)$, call it a combinatorial Hilbert space, namely the set of all supports of all vectors ...

**8**

votes

**2**answers

765 views

### failure of $\square(\kappa)$ at an inaccessible $\kappa$

How can we force the failure of $\square(\kappa)$ at an inaccessible $\kappa$, where
$\square(\kappa)$ is defined as follows: There is a sequence $(C_i:i< \kappa)$ such that:
(1) $C_{i+1} = \{i\}$...