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In 1970, J. Folkman proved that for a given integer r and a graph G of order n there exists a graph H with the same clique number as G such that every r coloring of vertices of H yields at least one monochromatic copy of G. His proof gives no good bound on the order of graph H, i.e. the order of H is bounded by an iterated power function. A related problem was studied by Łuczak, Ruciński and Urbański,...
The vertex Folkman number F(r,n,m), n<m, is the smallest integer t such that there exists a Km-free graph of order t with the property that every r-coloring of its vertices yields a monochromatic copy of Kn. The problem of bounding the Folkman numbers has been studied by several authors. However, in the most restrictive case, when m=n+1, no polynomial bound has been known for such numbers. In this...
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