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The work deals with combinatorial problems concerning colorings of non-uniform hypergraphs. Let $H=(V,E)$ be a hypergraph with minimum edge-cardinality $n$. We show that if $H$ is a simple hypergraph (i.e. every two distinct edges have at most one common vertex) and
$$\sum_{e\in E}r^{1-|e|}\leqslant c\sqrt{n},$$
for some absolute constant $c>0$, then $H$ is $r$-colorable. We also obtain a stronger result for triangle-free simple hypergraphs by proving that if $H$ is a simple triangle-free hypergraph and
$$\sum_{e\in E}r^{1-|e|}\leqslant c\cdot n,$$
for some absolute constant $c>0$, then $H$ is $r$-colorable.