Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the vertex Folkman numbers $F_v(a_1, ..., a_s; m - 1)$ when
  $\max\{a_1, ..., a_s\} = 6$ or $7$   Let $G$ be a graph and $a_1, ..., a_s$ be positive integers. Then $G
\overset{v}{\rightarrow} (a_1, ..., a_s)$ means that for every coloring of the
vertices of $G$ in $s$ colors there exists $i \in \{1, ..., s\}$, such that
there is a monochromatic $a_i$-clique of color $i$. The vertex Folkman number
$F_v(a_1, ..., a_s; q)$ is defined by the equality: $$ F_v(a_1, ..., a_s; q) =
\min\{|V(G)| : G \overset{v}{\rightarrow} (a_1, ..., a_s) \mbox{ and } K_q
\not\subseteq G\}. $$ Let $m = \sum\limits_{i=1}^s (a_i - 1) + 1$. It is easy
to see that $F_v(a_1, ..., a_s; q) = m$ if $q \geq m + 1$. In [11] it is proved
that $F_v(a_1, ..., a_s; m) = m + \max\{a_1, ..., a_s\}$. We know all the
numbers $F_v(a_1, ..., a_s; m - 1)$ when $\max\{a_1, ..., a_s\} \leq 5$ and
none of these numbers is known if $\max\{a_1, ..., a_s\} \geq 6$. In this paper
we compute the numbers $F_v(a_1, ..., a_s; m - 1)$ when $\max\{a_1, ..., a_s\}
= 6$.

19
The categories of this paper are: eess
37
The categories of this paper are: supr-con
32
The categories of this paper are: atom-ph
16
The categories of this paper are: stat
6
The categories of this paper are: astro-ph
22
The categories of this paper are: chao-dyn
30
The categories of this paper are: q-alg
35
The categories of this paper are: mtrl-th
0
The categories of this paper are: hep-ph
27
The categories of this paper are: comp-gas
10
The categories of this paper are: q-bio
29
The categories of this paper are: funct-an
7
The categories of this paper are: hep-th
12
The categories of this paper are: nucl-th
31
The categories of this paper are: ao-sci
33
The categories of this paper are: plasm-ph