Lagrangians of hypergraphs: The Frankl-Furedi conjecture holds almost everywhere

Frankl and Furedi conjectured in 1989 that the maximum Lagrangian of all r-uniform hypergraphs of fixed size m is realised by the initial segment of the colexicographic order. In particular, in the principal case m=their conjecture states that the maximum is attained on the clique of order t. We prove the latter statement for all r4 and large values of t (the case r=3 was settled by Talbot in 2002). More generally, we show for any r4 that the Frankl-Furedi conjecture holds whenever -rtr-2 for a constant r>0, thereby verifying it for most' mN. Furthermore, for r=3 we make an improvement on the results of Talbot and of Tang, Peng, Zhang and Zhao.