The triangle-free process

Abstract

Consider the following stochastic graph process. We begin with G 0 , the empty graph on n vertices, and form G i by adding a randomly chosen edge e i to G i−1 where e i is chosen uniformly at random from the collection of pairs of vertices that neither appear as edges in G i−1 nor form triangles when added as edges to G i−1 . Let the random variable M be the number of edges in the maximal triangle free graph generated by this process. We prove that asymptotically almost surely M = Θ(n 3/2 √ log n). This resolves a conjecture of Spencer. Furthermore, the independence number of G M is asymptotically almost surely Θ( √ n log n), which implies that the Ramsey number R(3, t) is bounded below by a constant times t 2 / log t (a fact that was previously established by Jeong Han Kim). The methods introduced here extend to the K 4 -free process, thereby establishing the bound R(4, t) = Ω(t 5/2 / log 2 t).

A note on the random greedy triangle-packing algorithm

“…The maximum 1-step difference is O(n 1/2 log 5/2 n) (as i < T implies bounds on the co-degrees). Thus the probability of such a large deviation beginning at step j is at most exp −Ω n 2 p(t(j)) 2 (n 2 p(t(j))) · n 1/2 log 5/2 n 2 = exp −Ω np(t(j)) log 5 n .…”

“…To bound the probability of a large deviation we recall a Lemma 6 from [2]. A sequence of random variables X 0 , X 1 , .…”

“…Now we turn to the lower bound on Q, namely (2). Here we work with the critical interval (7) q(t)n 3 − f (t) 2 n 2 log n p , q(t)n 3 − (f (t) − 1)f (t)n 2 log n p .…”

Maximum matchings in random bipartite graphs and the space utilization of Cuckoo Hash tables

“…One can allow for this in [24], but it is unsatisfactory to ask the reader to check this. We have decided to use an approach suggested in Bohman [4].…”

Paul Erdős and Probabilistic Reasoning

“…The detailed analysis is subtle, and is based on certain large deviation inequalities. An alternative analysis of this probabilistic construction, inspired by the differential equation method of Wormald [72], is given by Bohman in [17]. It is worth noting that the question of obtaining a super-linear lower bound for r(K 3 , K m ) is mentioned already in [26], and Erdős has established in [28], by an appropriate probabilistic construction, an Ω(m 2 / log 2 m) lower bound.…”