Lagrangians of hypergraphs: The Frankl-F\"uredi conjecture holds almost everywhere

Frankl and F"uredi conjectured in 1989 that the maximum Lagrangian of all $r$-uniform hypergraphs of fixed size $m$ is realised by the initial segment of the colexicographic order. In particular, in the principal case $m=\left(\genfrac{}{}{0px}{}{t}{r}\right)$ their conjecture states that every $H\subseteq $\mathbb{N}$^{®}$ofsize$\binom{t}{r}$ satisfies \begin{align*} \max {\sum_{A \in H}\prod_{i\in A} y_i \ \colon \ y_1,y_2,\ldots \geq 0; \sum_{i\in $\mathbb{N}$} y_i=1 }&\leq \frac{1}{t^r}\binom{t}{r}. \end{align*} We prove the above statement for all $r\ge 4$ and large values of $t$ (the case $r=3$ was settled by Talbot in 2002). More generally, we show for any $r\ge 4$ that the Frankl-F"uredi conjecture holds whenever $\binom{t-1}{r} \leq m \leq \binom{t}{r}- \gamma_r t^{r-2}$foraconstant$\gamma_r>0$, thereby verifying it for `most’ $m\in \mathbb{N}$. Furthermore, for $r=3$ we make an improvement on the results of Talbot~\cite{Tb} and Tang, Peng, Zhang and Zhao~\cite{TPZZ}.