Quasirandomness and Regularity: Lecture Notes I

The idea behind quasirandomness is that many random-like properties of combinatorial objects are equivalent, although they often do not appear to be so on first glance. We begin with the first class to which this notion was applied, and perhaps the most studied of combinatorial objects, graphs. It is easy to show that each of the following properties are true with probability 1 of a uniformly chosen random graph on n vertices, where n → ∞. (More precisely, to say that f(G) = o(h(n)) is to say that, for all > 0, there is an n0 so that n > n0 implies that f(G) < h(n) with probability at least 1− in G(n, 1/2).) It is far less obvious that deterministically these properties actually imply one another. We say that Q1 ⇒ Q2 if, whenever property Q1 says that f1(G) = o(h1(n)) and property Q2 says that f2(G) = o(h2(n)), then, for a sequence of graphs Gi on ni vertices with ni →∞, we have f1(Gni)/h1(ni)→ 0⇒ f2(Gni)/h2(ni)→ 0.