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Results tagged with infinite-combinatorics
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user 6085
Combinatorial properties of infinite sets. This is a corner-point of set theory and combinatorics.
5
votes
Accepted
Choosing *two* representatives
I assume by $\aleph$ you mean $\mathfrak c$, the cardinality of the continuum. You can build $D$ by transfinite recursion: Well-order the continuum in type $\mathfrak c$. At stage $\alpha$ you add a p …
- 32.5k
10
votes
Accepted
chain condition of a product of posets
First of all, note (as Monroe does in his question) that if $\mathbb P,\mathbb Q$ are ccc, then $\mathbb P\times\mathbb Q$ is $\mathfrak c^+$-cc, as an immediate consequence of the Erdős-Rado theorem …
- 32.5k
6
votes
Accepted
A problem about Ramsey Property
For $b=\omega+2$, see
András Hajnal. Some results and problems on set theory, Acta Math. Acad. Sci. Hungar., 11, (1960), 277–298. MR0150044 (27 #47).
In this paper, András shows that $\omega_1 …
- 32.5k
11
votes
Accepted
Does "$X \not\to (\omega)^\omega_2$ for every infinite $X$" imply ${\sf AC}$?
The answer is no, the statement that for every set $X$ we have $$X\not\to(\omega)^\omega_2$$ does not imply the axiom of choice.
This was shown by Kleinberg and Seiferas in 1973, see
MR0340025 (49 #4 …
- 32.5k