Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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
$m=\binom{t}{r}$ 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-F\"uredi 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}.

1
The categories of this paper are: cs
9
The categories of this paper are: nlin
36
The categories of this paper are: cmp-lg
33
The categories of this paper are: plasm-ph
7
The categories of this paper are: hep-th
31
The categories of this paper are: ao-sci
24
The categories of this paper are: dg-ga
19
The categories of this paper are: eess
10
The categories of this paper are: q-bio
5
The categories of this paper are: gr-qc
11
The categories of this paper are: hep-lat
14
The categories of this paper are: math-ph
32
The categories of this paper are: atom-ph
25
The categories of this paper are: patt-sol
22
The categories of this paper are: chao-dyn
18
The categories of this paper are: econ