Positional Games and the Second Moment Method

Dedicated to the memory of Paul Erdős

We study the fair Maker–Breaker graph Ramsey game MB(n;q). The board is , the players alternately occupy one edge a move, and Maker wants a clique of his own. We show that Maker has a winning strategy in MB(n;q) if , which is exactly the clique number of the random graph on n vertices with edge-probability 1/2. Due to an old theorem of Erdős and Selfridge this is best possible apart from an additive constant.