  A well-known conjecture of Frankl and F\"{u}redi from 1989 states that an
initial segment of colex of 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 and a counterexample to
an old conjecture of Ahlswede and Katona.
