On the 3-colouring vertex Folkman number \(F(2,2,4)\) (Q2752987)

For any fixed integers \(2\leq a_1\leq\ldots\leq a_r\), \(r\geq 2\), the (vertex) Folkman number \(F(a_1,\ldots,a_r)\) is defined as the minimal \(F\) such that there exists a graph \(G\) with \(F\) vertices and with the following properties: (i) \(G\) contains an \(a_r\)-clique (the complete graph with \(a_r\) vertices) and does not contain an \(a_r+1\)-clique. (ii) For every \(r\)-colouring of the vertices of \(G\) there exists an \(i=1,\ldots,r\), such that \(G\) contains a monochromatic \(a_i\)-clique of colour \(i\). NEWLINENEWLINENEWLINE\textit{J. Folkman} [SIAM J. Appl. Math. 18, 19-24 (1970; Zbl 0193.53103)] proved that \(F(a_1,\ldots,a_r)\) is finite for any \(a_1,\ldots,a_r\). The exact values of the Folkman numbers are known in few cases only. In particular, for colourings in more than two colours the only known values are \(F(2,2,2)=11\) [\textit{J. Mycielski}, Colloq. Math. 3, 161-162 (1955; Zbl 0064.17805) and \textit{V. Chvátal}, Lect. Notes Math. 406, 243-246 (1974; Zbl 0297.05107)] and \(F(2,2,2,2)=22\) [\textit{T. Jensen} and \textit{G. F. Royle}, J. Graph Theory 19, No. 1, 107-116 (1995; Zbl 0821.05023)]. The main result of the paper under review is that \(F(2,2,4)=13\). It is interesting to mention that the exact value of \(F(2,2,3)\) is still unknown.