When Does the Zero - One Law Hold ? Tomasz Luczak And

In 1960 Paul Erdos and Alfred Renyi [ER] began the subject of random graphs. The 1985 book of Bela Bollobas [B) provides the standard reference for this field. In modern terminology the random graph G(n, p) is a graph on vertex set [n] = {I, ... , n} where each pair i, j of vertices are adjacent with independent probability p. More accurately, G(n, p) is a probability space over the space of graphs on vertex set [n]. For any property A of graphs there is a probability, denoted Pr[G(n, p) FA], that G(n, p) satisfies A. In their very title, "On the evolution of random graphs," Erdos and Renyi envisioned a dynamic process, G( n , p) changing character as p moved from zero to one. They discovered (as did their many successors) that for many natural properties A Pr[G(n, p) F A] was usually near zero or near one and made the jump from near zero to near one (or back again) in a very narrow range. The placement of this critical range of p depended on n. For example, let A be the property of containing a triangle. There are m '" n3/6 potential triangles, each is a triangle in G(n, p) with probability p3, and so the expected number of triangles in G(n, p) is asymptotically n3p 3/6 . This suggests the critical range p = 8( 1 In). Indeed, Erdos and Renyi proved that if p = p(n) « lin then limn .... co Pr[G(n, p) F A] = 0, while if p = p(n) :» lin then limn .... co Pr[G(n, p) F A] = 1. (Notation: f(n)« g(n) means limn .... co f(n)1 g(n) = 0 while f(n) :» g(n) means limn .... co f(n)1 g(n) = +00.) They called p(n) = lin a threshold function for this property A. As other examples, connectivity has threshold functiop (log n) In, containing a clique on four points has threshold function n -2/3 , containing an edge has (easily!) threshold function n -2 , and every vertex lying in a triangle has threshold function (10gn)I/3 n-2/3. It was the observation that threshold functions seemed to be of the form (logntn-P with a, P rational that motivated our current line of research. What can we say about the possible threshold functions of properties A? Not much if we place no restrictions on A. For example, the property that the number of edges is even shows no threshold function behavior. If we restrict