On the number of edges in colour-critical graphs and hypergraphs

A (hyper)graph G is called k-critical if it has chromatic number k, but eve ry proper sub(hyper)graph of it is (k-1)-colourable. We prove that for suff iciently large k, every k-critical triangle-free graph on n vertices has at least (k-o(k))n edges. Furthermore, we show that every (k+1)-critical hype rgraph on n vertices and without graph edges has at least (k-3/3 rootk)n ed ges. Both bounds differ from the best possible bounds by o(kn) even for gra phs or hypergraphs of arbitrary girth.