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