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### 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 ...

**5**

votes

**1**answer

311 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 regressive functions
$f$ from $\omega_1$ minus singleton $0$ into $\omega_1$,
...

**6**

votes

**2**answers

652 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 ...

**6**

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**2**answers

323 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

191 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

247 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 ...

**6**

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**1**answer

206 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 ...

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**1**answer

135 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

148 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 $c$ ...

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**1**answer

402 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 ...

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**0**answers

359 views

### Hrushovski's Construction

Zilber expressed a conjecture for $\aleph_{1}$- categorical theories.(In 80s)
Zilber's Conjecture: The geometry of any $\aleph_{1}$- categorical structure is one of the following:
a) Trivial (No ...

**6**

votes

**1**answer

627 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

66 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

217 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 ...

**7**

votes

**1**answer

213 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

153 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 ...

**5**

votes

**0**answers

296 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 $(x_0, x_1, \dots)$ ...

**4**

votes

**2**answers

576 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

751 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 ...

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**5**answers

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### 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 ...

**10**

votes

**0**answers

222 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 ...

**7**

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663 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} = ...