Geodesic
On a Class of Vertex Folkman Numbers
Serdica Mathematical Journal, Tome 28 (2002) no. 3, pp. 219-232

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Let a1 , . . . , ar, be positive integers, i=1 ... r, m = ∑(ai − 1) + 1 and p = max{a1 , . . . , ar }. For a graph G the symbol G → (a1 , . . . , ar ) means that in every r-coloring of the vertices of G there exists a monochromatic ai -clique of color i for some i ∈ {1, . . . , r}. In this paper we consider the vertex Folkman numbers F (a1 , . . . , ar ; m − 1) = min |V (G)| : G → (a1 , . . . , ar ) and Km−1 ⊂ G} We prove that F (a1 , . . . , ar ; m − 1) = m + 6, if p = 3 and m ≧ 6 (Theorem 3) and F (a1 , . . . , ar ; m − 1) = m + 7, if p = 4 and m ≧ 6 (Theorem 4).
Keywords: Vertex Folkman Graph, Vertex Folkman Number 
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     title = {On a {Class} of {Vertex} {Folkman} {Numbers}},
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Nenov, Nedyalko. On a Class of Vertex Folkman Numbers. Serdica Mathematical Journal, Tome 28 (2002) no. 3, pp. 219-232. http://geodesic.mathdoc.fr/item/SMJ2_2002_28_3_a3/