New Upper Bound on Vertex Folkman Numbers

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, Rucinski and Urbanski, who gave some explicite bound on the order of H when G is a clique. In this note we give an alternative proof of Folkman's theorem with the relatively small order of H bounded from above by O(n3 log3 n). This improves Łuczak, Rucinski and Urbanski's result.