Some extremal properties concerning transitivity in graphs

I n this no te we consider only nont r iv ia l labelled o r i en t ed graphs , i.e. d ig raphs D hav i ng a t least one arc, no loops, and for each pa i r of points a a n d b of D a t mos t one of the arcs ab and ba is in D. D is transitive i f arc as is in D wheneve r arcs ab and bc are in D. We inves t iga te the n u m b e r of arcs of the largest t r ans i t i ve s u b g r a p h con ta ined in a ( round robin) t o u r n a m e n t , i.e. a comple te o r ien ted graph. Deno te by F(n ) the grea tes t in teger so t h a t eve ry t o u r n a m e n t on n points contains a ~ransi t ive subg raph of F(n) arcs. We will p rove