Avoidance of a giant component in half the edge set of a random graph

Let e1, e2, . . . be a sequence of edges chosen uniformly at random from the edge set of the complete graph Kn (i.e. we sample with replacement). Our goal is to choose, for m as large as possible, a subset E ⊆ {e1, e2, . . . , e2m}, |E| = m, such that the size of the largest component in G = ([n], E) is o(n) (i.e. G does not contain a giant component). Furthermore, the selection process must take place on-line; that is, we must choose to accept or reject an ei based on the previously seen edges e1, . . . , ei−1. We describe an on-line algorithm that succeeds whp for m = .9668n. Furthermore, we find a tight threshold for the off-line version of this question; that is, we find the threshold for the existence of m out of 2m random edges without a giant component. This threshold is m = cn where c satisfies a certain transcendental equation, c ∈ [.9792, .9793]. We also establish new upper bounds for more restricted Achlioptas processes.