# Tag Info

Accepted

### "Rocket elements" in bijections $f:\mathbb{N}\to \mathbb{N}$

Choose any infinite set $A\subseteq\mathbb N$ such that $\mu^+(A)=0$. Enumerate both $A$ and $B=\mathbb N\setminus A$ as \begin{align*} A&=\{a_1<a_2< \dots < a_n < \dots\}\\ B&=\{...
• 4,580
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• 5,047
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### Family of sets with a covering property

Yes. Let $S$ be a set of cardinality $|S|\ge k+2$. Let $X=\binom Sk$, the set of all $k$-element subsets of $S$. For each $s\in S$ let $a_s=\{x\in X:s\notin x\}$, and let $A=\{a_s:s\in S\}$. It is ...
• 9,221
Accepted

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

The following theorem of Kurepa answers the question. Theorem (Kurepa) Suppose that $\kappa$ is regular, $\lambda< \kappa$ , and $T$  is a $\kappa$-tree each of whose levels has cardinality ...
• 30.9k
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• 24.4k
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### "Drinking number" of a graph

Known as unfriendly partition conjecture. Open for countable graphs: http://www.openproblemgarden.org/op/unfriendly_partitions.
• 136

### Diagonalizing against $\omega_1$-sequences of functions mod finite

This isn't an answer, as you're working in ZFC. But it seems worth noting. Assume ZFC + AD$^{L(\mathbb{R})}$. Then $L(\mathbb{R})$ satisfies ZF + AD + DC + "the statement is false". Proof: ...
• 6,549
Accepted

### Non-isomorphic projective planes on $\omega$

You ask for the number of isomorphism classes of projective planes on $\omega$. I claim that it is exactly $2^{\aleph_0}$. It is at most $2^{\aleph_0}$. Indeed, a projective plane on $\omega$ can ...
• 25.4k

### Dominating families in bigger cardinals

Here is some general background information. The relevant search phrases for this topic are generalized cardinal invariants or generalized cardinal characteristics, and the topic has a growing ...
Accepted

### "König's theorem" for $T_2$-spaces?

What you are calling the "matching number" of $X$ is usually called its Souslin number -- the smallest cardinal bounding the size of any collection of pairwise disjoint open subsets of $X$. What you ...
• 16.3k
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### Cardinality of families of subsets of $\mathbb{N}$ whose intersections are finite

I guess this is a variant of Noah's construction: Let $S$ be an infinite subset of $\mathbf{N}=\{0,1,2,\dots\}$. Define a leafless rooted tree $V_S$, starting from a root at level 0, such that given ...
• 55.4k
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• 31.6k

### Dominating families in bigger cardinals

Professor Hamkins already gave many interesting references. Let me add a few more. Possibly, the work of Cummings-shelah Cardinal invariants above the continuum is the starting point for the study of ...
• 30.9k
Accepted

### Destroying Suslin, nothing special

Chapter IX of Proper and Improper Forcing addresses this issue. Shelah proves that Souslin's Hypothesis does not imply every Aronszajn tree is special, and he does this by investigating weak notions ...
• 7,172
Here is a counterexample. Consider $\mathbb{R}$ with addition. We define a topology on this group by giving the cosets of all countable index subgroups as a sub-basis. This subbasis is actually a ...