Abstract
The triangle-free process begins with an empty graph on n vertices and iteratively adds edges chosen uniformly at random subject to the constraint that no triangle is formed. We determine the asymptotic number of edges in the maximal triangle-free graph at which the triangle-free process terminates. We also bound the independence number of this graph, which gives an improved lower bound on Ramsey numbers: we show R(3,t) > (1/4 − o(1))t 2/ log t, which is within a 4 + o(1) factor of the best known upper bound. Furthermore, we determine which bounded size subgraphs are likely to appear in the maximal triangle-free graph produced by the triangle-free process: they are precisely those triangle-free graphs with maximal average density at most 2.
Research supported in part by NSF grants DMS-1001638 and DMS-1100215.
Research supported in part by ERC grant 239696 and EPSRC grant EP/G056730/1.
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Bohman, T., Keevash, P. (2013). Dynamic concentration of the triangle-free process. In: Nešetřil, J., Pellegrini, M. (eds) The Seventh European Conference on Combinatorics, Graph Theory and Applications. CRM Series, vol 16. Edizioni della Normale, Pisa. https://doi.org/10.1007/978-88-7642-475-5_78
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DOI: https://doi.org/10.1007/978-88-7642-475-5_78
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