Inventiones mathematicae

, Volume 181, Issue 2, pp 291–336 | Cite as

The early evolution of the H-free process

  • Tom Bohman
  • Peter Keevash


The H-free process, for some fixed graph H, is the random graph process defined by starting with an empty graph on n vertices and then adding edges one at a time, chosen uniformly at random subject to the constraint that no H subgraph is formed. Let G be the random maximal H-free graph obtained at the end of the process. When H is strictly 2-balanced, we show that for some c>0, with high probability as n→∞, the minimum degree in G is at least \(cn^{1-(v_{H}-2)/(e_{H}-1)}(\log n)^{1/(e_{H}-1)}\). This gives new lower bounds for the Turán numbers of certain bipartite graphs, such as the complete bipartite graphs K r,r with r≥5. When H is a complete graph K s with s≥5 we show that for some C>0, with high probability the independence number of G is at most \(Cn^{2/(s+1)}(\log n)^{1-1/(e_{H}-1)}\). This gives new lower bounds for Ramsey numbers R(s,t) for fixed s≥5 and t large. We also obtain new bounds for the independence number of G for other graphs H, including the case when H is a cycle. Our proofs use the differential equations method for random graph processes to analyse the evolution of the process, and give further information about the structure of the graphs obtained, including asymptotic formulae for a broad class of subgraph extension variables.


Random Graph Minimum Degree Extension Variable Free Graph Independence Number 
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© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Department of Mathematical SciencesCarnegie Mellon UniversityPittsburghUSA
  2. 2.School of Mathematical SciencesQueen Mary, University of LondonLondonUK

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