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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| Veröffentlicht in: | Random structures & algorithms Jg. 58; H. 2; S. 221 - 293 |
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| Hauptverfasser: | , |
| Format: | Journal Article |
| Sprache: | Englisch |
| Veröffentlicht: |
New York
John Wiley & Sons, Inc
01.03.2021
Wiley Subscription Services, Inc |
| Schlagworte: | |
| ISSN: | 1042-9832, 1098-2418 |
| Online-Zugang: | Volltext |
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| Zusammenfassung: | 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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| Bibliographie: | 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 |
| ISSN: | 1042-9832 1098-2418 |
| DOI: | 10.1002/rsa.20973 |