  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 $\max\{a_1, ..., a_s\}
= 6$.
