The vertex Folkman Numbers \(F_{v}(a_{1}, …,a_{s}; m-1) =m+9,\) if \(\max\{a_{j},…,a_{s}\} =5\)

For a graph \( G \), the expression \( G \overset{v}{\rightarrow} (a_1,\ldots,a_s) \) means that for any \( s \)-coloring of the vertices of \( G \), there exists \( i \in \{1,\ldots,s\} \) such that there is a monochromatic \( a_i \)-clique of color \( i \). The vertex Folkman numbers

\[ F_v(a_1,\ldots,a_s;m-1) = \min\{|V(G)|: G \overset{v}{\rightarrow} (a_1,\ldots,a_s) \text{ and } K_{m-1} \nsubseteq G\} \]

are considered, where \( m = \sum_{i=1}^s (a_i – 1) + 1 \).

With the help of a computer, we show that \( F_v(2,2,5;6) = 16 \), and then we prove

\[ F_v(a_1,\ldots,a_s;m-1) = m+9, \]

if \( \max\{a_1,\ldots,a_s\} = 5 \).

We also obtain the bounds

\[ m+9 \leq F_v(a_1,\ldots,a_s;m-1) \leq m+10, \]

if \( \max\{a_1,\ldots,a_s\} = 6 \).