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

Abstract

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 $2^{k+\widetilde{\Theta}(k^{1/2})}$.