On the construction of 3-chromatic hypergraphs with few edges

The minimum number m ( n ) of edges in a 3-chromatic n-uniform hypergraph has been widely studied in the literature. The best known upper bound is due to Erdős, who showed, using the probabilistic method, that m ( n ) ⩽ O ( n 2 2 n ) . Abbott and Moser gave an explicit construction of a 3-chromatic n-uniform hypergraph with at most ( 7 ) n ≈ 2.65 n hyperedges, which is the best known constructive upper bound. In this paper we improve this bound to 2 ( 1 + o ( 1 ) ) n . Our technique can also be used to describe n-uniform hypergraphs with chromatic number at least r + 1 and at most r ( 1 + o ( 1 ) ) n hyperedges, for every r ⩾ 3 .