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Dynamic concentration of the triangle‐free process
Author(s) -
Bohman Tom,
Keevash Peter
Publication year - 2021
Publication title -
random structures and algorithms
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.314
H-Index - 69
eISSN - 1098-2418
pISSN - 1042-9832
DOI - 10.1002/rsa.20973
Subject(s) - combinatorics , mathematics , upper and lower bounds , bounded function , graph , discrete mathematics , constraint (computer aided design) , random regular graph , chordal graph , 1 planar graph , geometry , mathematical analysis
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 ) ) t 2 / log t , 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.