Lagrangians of Hypergraphs

How large can the Lagrangian of an r-graph with m edges be? Frankl and Füredi [1] conjectured that the r-graph of size m formed by taking the first m sets in the colex ordering of N(r) has the largest Lagrangian of all r-graphs of size m. We prove the first ‘interesting’ case of this conjecture, namely that the 3-graph with (t3) edges and largest Lagrangian is [t](3). We also prove that this conjecture is true for 3-graphs of several other sizes.For general r-graphs we prove a weaker result: for t sufficiently large, the r-graph of size (tr) supported on t + 1 vertices and with largest Lagrangian, is [t](r).