Frankl–Füredi conjecture on hypergraph Lagrangians

For a set $X$, let $X^{(r)}$ denote the family of all subsets of $X$ of size $r$. An $r$-graph is a family $G\subseteq\mathbb{N}^{(r)}$, and its Lagrangian is

$$\lambda(G):=\max\left\{\sum_{e\in E(G)}\prod_{x\in e}w(x): w(x)\geq 0\text{ for all }x\in\mathbb{N},\ \sum_{x\in\mathbb{N}}w(x)=1\right\}.$$

Let $\mathcal{C}(m,r)$ be the family consisting of the first $m$ members of the colexicographic order on $\mathbb{N}^{(r)}$, where $A<B$ if and only if $\sum_{i\in A}2^i<\sum_{i\in B}2^i$.

Frankl–Füredi conjecture. If $G$ is an $r$-graph with $m$ edges, then

$$\lambda(G)\leq\lambda(\mathcal{C}(m,r)).$$

This conjecture asks which $r$-graph with a prescribed number of edges maximizes the Lagrangian. It is a central problem in extremal combinatorics and is false in general, as indicated by the paper's title; the conjecture is therefore refuted, although particular cases may still hold.