Bart–Moe games, JumbleG and discrepancy

Let A and B be hypergraphs with a common vertex set V . In a ( p , q , A ∪ B ) Bart–Moe game, the players take turns selecting previously unclaimed vertices of V . The game ends when every vertex has been claimed by one of the players. The first player, called Bart (to denote his role as Breaker and Avoider together), selects p vertices per move and the second player, called Moe (to denote his role as Maker or Enforcer), selects q vertices per move. Bart wins the game iff he has at least one vertex in every hyperedge B ∈ B and no complete hyperedge A ∈ A . We prove a sufficient condition for Bart to win the ( p , 1 ) game, for every positive integer p . We then apply this criterion to two different games in which the first player’s aim is to build a pseudo-random graph of density p p + 1 , and to a discrepancy game.