Computation of the vertex Folkman numbers \(F(2,2,2,4;6)\) and \(F(2,3,4;6)\) (Q5960784)

Two vertex Folkman numbers are determined, namely \(F(2,2,2,4;6)=14\) and \(F(2,3,4;6)=14\). The vertex Folkman number \(F(a_1,\ldots,a_r;q)\) is defined as the smallest integer \(F\) such that there exists a graph \(G\) with \(F\) vertices and clique number less than \(q\) such that every \(r\)-coloring of the vertices of \(G\) contains at least one monochromatic \(a_i\)-clique of some color \(i\). The proof of the upper bound utilizes a graph constructed by \textit{R. E. Greenwood} and \textit{A. M. Gleason} [Can. J. Math. 7, 1-7 (1955; Zbl 0064.17901)] to show that the Ramsey number \(R(3,5) \geq 14\).