New answers tagged ramsey-theory
2
votes
Is there an uncountable extension of the Ramsey set $[\omega]^2$?
If $N$ is an infinite proper subset of $\omega$ then the family $\mathcal A=[\omega]^2\cup\{A\subseteq\omega:A\not\subseteq N\}$ is a Ramsey family and $|\mathcal A|=2^{\aleph_0}$.
On the other hand, ...
- 13.4k
4
votes
Accepted
Is there an uncountable extension of the Ramsey set $[\omega]^2$?
Yes. Just take $\mathcal{A}$ to be $[\omega]^2$ together with the powerset of the even integers.
- 32.1k
9
votes
Accepted
Maximal Ramsey families
No. Consider the family $\mathcal{R}$ of all sets of size 1. It is certainly a Ramsey family. Consider a Ramsey extension $\mathcal{R}_1\supset \mathcal{R}$. Take a map $f$ which maps all sets from $\...
- 109k
4
votes
Accepted
Non-Ramsey function $f:[\omega]^{<\omega}\to\{0,1\}$
For $P \subset \omega$ finite, let $f(P) = \#P \bmod{2}$, i.e., $f(P) = 0$ if $P$ has even cardinal, and $1$ otherwise.
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