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The Final Size of the C4-Free Process


We consider the following random graph process: starting with n isolated vertices, add edges uniformly at random provided no such edge creates a copy of C4. We show that, with probability tending to 1 as n → ∞, the final graph produced by this process has maximum degree O((nlogn)1/3) and consequently size O(n4/3(logn)1/3), which are sharp up to constants. This confirms conjectures of Bohman and Keevash and of Osthus and Taraz, and improves upon previous bounds due to Bollobás and Riordan and Osthus and Taraz.

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[1] N. Alon , M. Krivelevich and B. Sudakov (1999) Coloring graphs with sparse neighborhoods. J. Combin. Theory Ser. B 77 7382.

[2] T. Bohman (2009) The triangle-free process. Adv. Math. 221 16531677.

[3] T. Bohman and P. Keevash (2010) The early evolution of the H-free process. Invent. Math. 181 291336.

[6] P. Erdős , S. Suen and P. Winkler (1995) On the size of a random maximal graph. Random Struct. Alg. 6 309318.

[8] J. H. Kim (1995) The Ramsey number R(3, t) has order of magnitude t2/logt. Random Struct. Alg. 7 173207.

[9] D. Osthus and A. Taraz (2001) Random maximal H-free graphs. Random Struct. Alg. 18 6182.

[12] J. Spencer (1990) Counting extensions. J. Combin. Theory Ser. A 55 247255.

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Combinatorics, Probability and Computing
  • ISSN: 0963-5483
  • EISSN: 1469-2163
  • URL: /core/journals/combinatorics-probability-and-computing
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