Lecture 2: the Probabilistic Method and Linearity of Expectation

An old math puzzle goes: Suppose there are six people in a room; some of them shake hands. Prove that there are at least three people who all shook each others’ hands or three people such that no pair of them shook hands. Generalized a bit, this is the classic Ramsey problem. The diagonal Ramsey numbers R(k) are de ned as follows. R(k) is the smallest integer n such that in every two-coloring of the edges of the complete graph Kn by red and blue, there is a monochromatic copy of Kk , i.e. there are k nodes such that all of the k 2 edges between them are red or all of the edges are blue. A solution to the puzzle above asserts that R(3) 6 6 (and it is easy to check that, in fact, R(3) 6). In 1929, Ramsey proved that R(k) is nite for every k. We want to show that R(k) must grow pretty fast; in fact, we’ll prove that for k > 3, we have R(k) > b2k/2c. This requires nding a coloring of Kn that doesn’t contain any monochromatic Kk . To do this, we’ll use the probabilistic method: We’ll give a random coloring of Kn and show that it satis es our desired property with positive probability. This proof appeared in a paper of Erdös from 1947, and this is the example that starts Alon and Spencer’s famous book devoted to the probabilistic method.