New Upper Bound for a Class of Vertex Folkman Numbers

@article{Kolev2006NewUB,
  title={New Upper Bound for a Class of Vertex Folkman Numbers},
  author={Nikolay Rangelov Kolev and Nedyalko Dimov Nenov},
  journal={Electr. J. Comb.},
  year={2006},
  volume={13}
}
Let a1 ,. . . , ar be positive integers, m = ∑r i=1(ai−1)+1 and p = max{a1, . . . , ar}. For a graph G the symbol G → {a1, . . . , ar} denotes that in every r-coloring of the vertices of G there exists a monochromatic ai-clique of color i for some i = 1, . . . , r. The vertex Folkman numbers F (a1, . . . , ar;m − 1) = min{|V (G)| : G → (a1 . . . ar) and Km−1 6⊆ G} are considered. We prove that F (a1, . . . , ar;m − 1) ≤ m + 3p, p ≥ 3. This inequality improves the bound for these numbers… CONTINUE READING

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