Abstract
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, 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(n 3log3 n). This improves Łuczak, Ruciński and Urbański’s result.
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Dudek, A., Rödl, V. (2008). New Upper Bound on Vertex Folkman Numbers. In: Laber, E.S., Bornstein, C., Nogueira, L.T., Faria, L. (eds) LATIN 2008: Theoretical Informatics. LATIN 2008. Lecture Notes in Computer Science, vol 4957. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-78773-0_41
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DOI: https://doi.org/10.1007/978-3-540-78773-0_41
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