Dynamic concentration of the triangle‐free process
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....
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| Vydáno v: | Random structures & algorithms Ročník 58; číslo 2; s. 221 - 293 |
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| Médium: | Journal Article |
| Jazyk: | angličtina |
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John Wiley & Sons, Inc
01.03.2021
Wiley Subscription Services, Inc |
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| ISSN: | 1042-9832, 1098-2418 |
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| 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 the Ramsey numbers R(3, t): we show R(3,t)>(1/4−o(1))t2/logt, which is within a 4 + o(1) factor of the best known upper bound. Our improvement on previous analyses of this process exploits the self‐correcting nature of key statistics of the process. 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 density at most 2. |
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| AbstractList | 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 the Ramsey numbers R(3, t): we show R(3,t)>(1/4−o(1))t2/logt, which is within a 4 + o(1) factor of the best known upper bound. Our improvement on previous analyses of this process exploits the self‐correcting nature of key statistics of the process. 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 density at most 2. 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 the Ramsey numbers R (3, t ): we show , which is within a 4 + o (1) factor of the best known upper bound. Our improvement on previous analyses of this process exploits the self‐correcting nature of key statistics of the process. 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 density at most 2. 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 the Ramsey numbers R(3, t): we show R(3,t)>(1/4−o(1))t2/logt, which is within a 4 + o(1) factor of the best known upper bound. Our improvement on previous analyses of this process exploits the self‐correcting nature of key statistics of the process. 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 density at most 2. |
| Author | Keevash, Peter Bohman, Tom |
| Author_xml | – sequence: 1 givenname: Tom surname: Bohman fullname: Bohman, Tom email: tbohman@math.cmu.edu organization: Carnegie Mellon University – sequence: 2 givenname: Peter surname: Keevash fullname: Keevash, Peter organization: University of Oxford |
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| Cites_doi | 10.1214/aop/1176996452 10.1016/j.aim.2009.02.018 10.1002/rsa.20444 10.1017/S0963548311000368 10.1016/j.aim.2015.04.015 10.1007/s00222-010-0247-x 10.37236/1496 10.1002/rsa.3240060217 10.1002/1098-2418(200101)18:1<61::AID-RSA5>3.0.CO;2-T 10.1002/rsa.20378 10.1002/jgt.20453 10.37236/93 10.1137/110824097 10.1007/s004930070014 10.37236/655 10.37236/1192 10.1002/0471722154 10.1016/0012-365X(83)90273-X 10.1017/CBO9781316339831 10.4153/CJM-1961-029-9 10.4310/JOC.2010.v1.n4.a5 10.1017/S0963548300000183 10.1002/rsa.20468 10.1016/0012-365X(77)90044-9 10.1016/0097-3165(90)90070-D 10.1002/rsa.20517 10.1016/j.disc.2011.08.008 10.1002/rsa.3240070302 10.1016/0097-3165(80)90030-8 |
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| Notes | Funding information This research was supported by the NSF Grants, DMS‐1001638 and DMS‐1100215 (T.B.). ERC Grants, 647678 and 239696. EPSRC Grant, EP/G056730/1 (P.K.). ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
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| Snippet | 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... 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... |
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| SubjectTerms | Apexes dynamic concentration Graph theory Lower bounds Ramsey numbers triangle‐free Upper bounds |
| Title | Dynamic concentration of the triangle‐free process |
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