Lagrangians of hypergraphs II: When colex is best

A well-known conjecture of Frankl and Furedi from 1989 states that an initial segment of the colexicographic order has the largest Lagrangian of any r-uniform hypergraph with m hyperedges. We show that this is true when r = 3. We also give a new proof of a related conjecture of Nikiforov for large t and a counterexample to an old conjecture of Ahlswede and Katona.