Dynamic concentration of the triangle-free process

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))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 2-density at most 2.

The Ramsey number of loose cycles versus cliques

“…It is known that The upper bound is due to Ajtai, Komlós, and Szemerédi , and the lower bound was obtained by Kim . The best constants implicit in are due to Shearer for the upper bound, and to Bohman and Keevash and Fiz Pontiveros et al for the lower bound (both groups simultaneouly obtained very similar results using the triangle‐free process).…”

Balancing sums of random vectors

“…To prove Theorem 2.4, we shall show that the best-of-two rule enforces 'self-correction'. Similar methods based on self-correction have recently been used to answer some long-standing questions about random graph processes; see [9,12,10], for example. This paper is organised as follows.…”

Hypergraph Matchings and Designs

“…The first goes back to the 'nibble' (semi-random) method of Rödl [90], introduced to solve the Erdős-Hanani [26] conjecture on approximate Steiner systems (see the next section), which has since had a great impact on Combinatorics (e.g. [5,10,11,12,13,14,28,29,38,50,51,66,69,76,85,99,104]). A special case of a theorem of Kahn [51] is that if there is a fractional perfect matching on the edges of an r-graph G on [n] such that for any pair of vertices x, y the total weight on edges containing {x, y} is o(1) then G has a matching covering all but o(n) vertices.…”

“…A similar idea appears in the Rödl-Schacht proof of the hypergraph regularity lemma via regular approximation (see [93]). It may also be viewed as a 'guided version' of the self-correction that appears naturally in random greedy algorithms (see [14,28]).…”