A phase transition for avoiding a giant component

Let c be a constant and (e1, f1), (e2, f2), . . . , (ecn, fcn) be a sequence of ordered pairs of edges from the complete graph Kn chosen uniformly and independently at random. We prove that there exists a constant c2 such that if c > c2 then whp every graph which contains at least one edge from each ordered pair (ei, fi) has a component of size Ω(n) and if c < c2 then whp there is a graph containing at least one edge from each pair that has no component with more than O ( n1−2 ) vertices, where 2 is a constant that depends on c2− c. The constant c2 is roughly 0.97677.