It is played by two players, called Waiter and Client.
In a Waiter-Client game, Waiter wins if he manages to occupy all the elements of a winning-set, while Client wins if he manages to prevent this, i.e., hold at least one element in each winning-set.
So Waiter and Client have, respectively, the same goals as Maker and Breaker in a Maker-Breaker game; only the rules for taking elements are different.
In some cases, the Waiter is much more powerful than the player with the same goal in the Maker-Breaker variant.
Then, when the board is infinite, Waiter can win as Maker for any k >= 1.
Moreover, even when the board is finite, Waiter always wins as Breaker when k >= 8.
[4] This leads to the following conjecture by József Beck:[2] If Maker wins the Maker-Breaker game on
Suppose the winning-sets are all of size k (i.e., the game-hypergraph is k-uniform).
In a Maker-Breaker game, the Erdos-Selfridge theorem implies that Breaker wins if the number of winning-sets is less than
[5] By the above conjecture, we would expect the same to hold in the corresponding Client-Waiter game - Waiter "should" win (as Breaker) whenever the number of winning-sets is less than