Can You Feel the Double Jump?

In their fundamental work Paul Erdős and Alfred Renyi [1] considered the evolution of the random graphG(n, p) as p “evolved” from 0 to 1. At p = 1/n a sudden and dramatic change takes place in G. When p = c/n with c < 1 the random G consists of small components, the largest of size Θ(log n). But by p = c/n with c > 1 many of the components have “congealed” into a “giant component” of size Θ(n). Erdős and Renyi called this the double jump, the terms phase transition (from the analogy to percolation) and Big Bang have also been proferred. Now imagine an observer who can only see G through a logical fog. He may refer to graph theoretic properties A within a limited logical language. Will he be able to detect the double jump? The answer depends on the strength of the language. Our rough answer to this rough question is: the double jump is not detectible in the First Order Theory of Graphs but it is detectible in the Second Order Monadic Theory of Graphs. These theories will be described below. We use the abbreviations fotog and somtog for these two theories respectively. For any property A and any c > 0 we define