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By N. Biggs

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Hence, for k fixed, we may couple the exploration above with the branching process X(c) so that the probability of a mismatch before both processes have reached size k + 1 is o(1). Let Nk (G) denote the number of vertices of a graph G in components of order k. The trivial argument above gives the following lemma. 42 B. Bollob´ as and O. Riordan Lemma 1. Let c > 0 and a positive integer k be fixed, and let Gn = G(n, c/n). Then 1 E Nk (Gn ) → P ( X(c) = k ), n where X(c) denotes the total number of particles in all generations of X(c).

Historical remarks As we have already mentioned, the greatest single result of Erd˝ os and R´enyi about random graphs concerns the ‘phase transition’ in the component structure of a random graph process or, equivalently, a random graph G(n, c/n), as c grows from c < 1 to c > 1. In fact, in [94] this is not so much a single theorem as an attractive formulation of the union of several results. We shall spend some time on the rather peculiar history of this result and the refinements that came many years later.

Lemma 10. For any 0 < p < 1, the binomial distribution Bi (n, p) is stochastically dominated by the Poisson distribution with mean −n log (1 − p) = np + O(np2 ). For any 0 < a < 3/2, the distribution Bi (n, a/n) stochastically dominates the Poisson distribution with mean b up to o n−100 , where b = a(1 − log n/n). Proof. For any p we may couple the (rather simple) binomial (or Bernoulli) distribution Bi (1, p) with the Poisson distribution with mean − log (1 − p) so that the latter dominates. To see this, note that if X and Y have these distributions, then P (X = 0) = 1 − p = P (Y = 0), while if Y = 0 then Y ≥ 1 ≥ X.

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