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

TL;DR

This paper improves the deterministic construction of 3-chromatic hypergraphs with fewer edges, narrowing the gap between known bounds and advancing understanding of hypergraph colorability.

Contribution

The authors derandomize Gebauer's construction, reducing the number of edges needed for 3-chromatic hypergraphs from exponential in k to a smaller exponential bound.

Findings

Reduced the number of edges in 3-chromatic hypergraphs

Applied derandomization techniques to existing constructions

Achieved tighter bounds on hypergraph edge counts

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