| Izvleček: | Frankl and Füredi 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=tr their conjecture states that the maximum is attained on the clique of order t. We prove the latter statement for all r≥4 and large values of t (the case r=3 was settled by Talbot in 2002). More generally, we show for any r≥4 that the Frankl-Füredi conjecture holds whenever t-1r≤m≤tr-γrtr-2 for a constant γr>0, thereby verifying it for 'most' m∈N. Furthermore, for r=3 we make an improvement on the results of Talbot and of Tang, Peng, Zhang and Zhao.
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