On small non-uniform hypergraphs without property B

For a given hypergraph $H=(V,E)$ consider the sum $q(H)$ of $2^{-\mathrm{\mid }e\mathrm{\mid }}$ over $e\in E$. Consider the class of hypergraphs with the smallest edge of size $n$ and without a 2-colouring without monochromatic edges. Let $q(n)$ be the smallest value of $q(H)$ in this class. We provide a survey of the known bounds on $q(n)$ and make some minor refinements.