Title: Lagrangians of hypergraphs: The Frankl-Füredi conjecture holds almost everywhere,
Abstract: Frankl and Füredi 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
$m=\binom{t}{r}$ their conjecture states that every $H\subseteq
\mathbb{N}^{(r)}$ of size $\binom{t}{r}$ 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 $r\geq 4$ and large values of $t$ (the
case $r=3$ was settled by Talbot in 2002). More generally, we show for any
$r\geq 4$ that the Frankl-Füredi conjecture holds whenever $\binom{t-1}{r}
\leq m \leq \binom{t}{r}- \gamma_r t^{r-2}$ for a constant $\gamma_r>0$,
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}.