Improving Gebauer's construction of 3-chromatic hypergraphs with few edges

In 1964 Erd\H{o}s proved, by randomized construction, that the minimum number of edges in a $k$-graph that is not two colorable is $O(k^2\; 2^k)$. To this day, it is not known whether there exist such $k$-graphs with smaller number of edges. Known deterministic constructions use much larger number of edges. The most recent one by Gebauer requires $2^{{k+\Theta(k{2/3})}$} edges. Applying derandomization technique we reduce that number to $2^{{k+\widetilde{\Theta}(k{1/2})}$.}