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8 votes

Uniformization of almost disjoint families

No, this is not consistent: there is (provably in ZFC) an almost disjoint family of size $\aleph_1$ and a two-valued function on that family such that the function cannot be uniformized in the way you'...
Will Brian's user avatar
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4 votes

Does $\mathbb{Z}\times\mathbb{Z}$ have an aperiodic monotile?

Yes. A $2$-by-$2$ square $\{0,1\}^2$ can tile $\mathbb{Z}^2$ with just one period. So $\{0,2\}^2$ can tile $2\mathbb{Z}^2 \leq \mathbb{Z}^2$ with just one period. Break other periods in the other ...
Ville Salo's user avatar
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2 votes

Is the chromatic number of hypergraphs downward continuous?

Fred Galvin had conjectured that the answer is "yes" for graphs in [1] (conjecture 2), in his paper he showed that the variation of the problem to induced graphs is consistently false: ...
Holo's user avatar
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1 vote

Order-embeddability of ${\frak b}$ and ${\frak d}$ in $\mathbb{R}$

Assume that $i : \alpha \to \mathbb{R}$ is an order-embedding of some ordinal $\alpha$ into $(\mathbb{R},<)$. We can modify $i$ to yield an order embedding $j : \alpha \to \mathbb{Q}$ by induction ...
Arno's user avatar
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