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on Erdős #1177 · set theory / chromatic number

The statement

Let $G$ be a finite $3$ -uniform hypergraph, and let $F_G(\kappa)$ denote the collection of $3$ -uniform hypergraphs with chromatic number $\kappa$ not containing $G$ .

If $F_G(\aleph_1)$ is not empty then there exists $X\in F_G(\aleph_1)$ of cardinality at most $2^{2^{\aleph_0}}$ .

If both $F_G(\aleph_1)$ and $F_H(\aleph_1)$ are non-empty then $F_G(\aleph_1)\cap F_H(\aleph_1)$ is non-empty.

If $\kappa,\lambda$ are uncountable cardinals and $F_G(\kappa)$ is non-empty then $F_G(\lambda)$ is non-empty.

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.