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Let $H=(V,E)$ be a hypergraph such that for every $e\in E$ we have $|e|\geq 2$. A map $c:V\to \kappa$, where $\kappa$ is a cardinal, is said to be a (hypergraph) coloring if for all $e\in E$ the restriction $c|_e$ is not constant.

Is there a hypergraph $H=(V,E)$ such that $|V|$ is infinite, $|e| = |V|$ for all $e\in E$ and for every cardinal $\kappa$ and coloring map $c:V\to \kappa$ we have "small fibers" in the sense that $$|c^{-1}(\{\alpha\})|<|V| \text{ for all } \alpha\in\kappa$$?

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Let $V$ be arbitrary and take $E$ to be the set of all subsets of $V$ of the same cardinality as $V$. If any coloring had a fiber of size $|V|$, then that fiber would be a monochromatic edge.

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