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.