# Hypergraph colorings with small fibers

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 $$|c^{-1}(\{\alpha\})|<|V| \text{ for all } \alpha\in\kappa$$?

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.