Hypergraph Lagrangians I: the Frankl-F\"uredi conjecture is false

An old and well-known conjecture of Frankl and F"{u}redi states that the Lagrangian of an $r$-uniform hypergraph with $m$ edges is maximised by an initial segment of colex. In this paper we disprove this conjecture by finding an infinite family of counterexamples for all $r\ge 4$. We also show that, for sufficiently large $t\in \mathbb{N}$, the conjecture is true in the range $\left(\genfrac{}{}{0px}{}{t}{r}\right)\le m\le \left(\genfrac{}{}{0px}{}{t+1}{r}\right)-\left(\genfrac{}{}{0px}{}{t-1}{r-2}\right)$.