Abstract

We discuss a branch of Ramsey theory concerning vertex Folkman numbers and how computer algorithms have been used to compute a new Folkman number. We write G ! (a1, . . . , ak)v if for every vertex k-coloring of an undirected simple graph G, a monochromatic Kai is forced in color i 2 {1, . . . , k}. The vertex Folkman number is defined as Fv(a1, . . . , ak; p) = min{|V (G)| : G ! (a1, . . . , ak)v ^ Kp 6 G}. Folkman showed in 1970 that this number exists for p > max{a1, . . . , ak}. Let m = 1+Pk i=1(ai−1) and a = max{a1, . . . , ak}, then Fv(a1, . . . , ak; p) = m for p > m, and Fv(a1, . . . , ak; p) = a +m for p = m. For p < m the situation is more difficult and much less is known. We show here that, for a case of p = m−1, Fv(2, 2, 3; 4) = 14.

Publication Date

2006

Comments

Note: imported from RIT’s Digital Media Library running on DSpace to RIT Scholar Works in February 2014.

Document Type

Article

Department, Program, or Center

Center for Advancing the Study of CyberInfrastructure

Campus

RIT – Main Campus

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