, Volume 37, Issue 4, pp 767–784 | Cite as

An exponential-type upper bound for Folkman numbers

  • Vojtěch Rödl
  • Andrzej Ruciński
  • Mathias Schacht
Original paper


For given integers k and r, the Folkman number f(k;r) is the smallest number of vertices in a graph G which contains no clique on k+1 vertices, yet for every partition of its edges into r parts, some part contains a clique of order k. The existence (finiteness) of Folkman numbers was established by Folkman (1970) for r=2 and by Nešetřil and Rödl (1976) for arbitrary r, but these proofs led to very weak upper bounds on f(k;r).

Recently, Conlon and Gowers and independently the authors obtained a doubly exponential bound on f(k;2). Here, we establish a further improvement by showing an upper bound on f(k;r) which is exponential in a polynomial of k and r. This is comparable to the known lower bound 2 Ω(rk). Our proof relies on a recent result of Saxton and Thomason (or, alternatively, on a recent result of Balogh, Morris, and Samotij) from which we deduce a quantitative version of Ramsey’s theorem in random graphs.

Mathematics Subject Classification (2000)

05D10 05C80 


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Copyright information

© János Bolyai Mathematical Society and Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  • Vojtěch Rödl
    • 1
  • Andrzej Ruciński
    • 2
  • Mathias Schacht
    • 3
  1. 1.Emory UniversityAtlantaUSA
  2. 2.A. Mickiewicz UniversityPoznańPoland
  3. 3.Universität HamburgHamburgGermany

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