Manipulative waiters with probabilistic intuition

For positive integers $n$ and $q$ and a monotone graph property $\cA$, we consider the two player, perfect information game $\WC(n,q,\cA)$, which is defined as follows. The game proceeds in rounds. In each round, the first player, called Waiter, offers the second player, called Client, $q+1$ edges of the complete graph $K_n$ which have not been offered previously. Client then chooses one of these edges which he keeps and the remaining $q$ edges go back to Waiter. If at the end of the game, the graph which consists of the edges chosen by Client satisfies the property $\cA$, then Waiter is declared the winner; otherwise Client wins the game. In this paper we study such games (also known as Picker-Chooser games) for a variety of natural graph theoretic parameters, such as the size of a largest component or the length of a longest cycle. In particular, we describe a phase transition type phenomenon which occurs when the parameter $q$ is close to $n$ and is reminiscent of phase transition phenomena in random graphs. Namely, we prove that if $q \leq (1 - \varepsilon) n$, then Client can avoid connected components of order $c \varepsilon^{-2} \ln n$ for some absolute constant $c > 0$, whereas, for $q \geq (1 + \varepsilon) n$, Waiter can force a giant, linearly sized, connected component in Client's graph. We also prove that Waiter can force Client's graph to be pancyclic for every $q \leq c n$, where $c > 0$ is an appropriate constant.