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Results tagged with infinite-combinatorics
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user 160347
Combinatorial properties of infinite sets. This is a corner-point of set theory and combinatorics.
6
votes
Accepted
Is the definable club filter normal?
Work in ZF. As Asaf was probably mentioning in his comment, you have asked two distinct questions, one in the title (referring to a filter), and one in the main body of the question (just referring to …
- 9,902
6
votes
Accepted
Jónsson functions for arbitrary ordinals
Here is an argument for the equivalence. It's easier in the case that $\kappa=\mathrm{card}(\gamma)$ is a regular cardinal, so let's consider that first. Fix a J'onsson function $f$ for $\kappa$. Fix …
- 9,902
12
votes
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: Work in $L( …
- 9,902
5
votes
Set sizes in linear set systems on $\mathbb{N}$ containing some disjoint sets
Yes. Fix a partition $\left<A_n\right>_{n\leq\omega}$
of $\mathbb{N}$ such that $A_n$ is infinite for each $n<\omega$, and $A_\omega$ has cardinality 3.
For $n<\omega$ let $B_\ell=\bigcup_{n\leq\ell\ …
- 9,902
8
votes
Accepted
Image-catching families in $\omega$
Original partial answer:
Here is some information:
An easy diagonalization shows that every image-catching family is uncountable. And more generally, MA implies that every image-catching family has ca …
- 9,902