Generalized Harary Games

There are a number of positional games known on the infinite chessboard. One of the most studied is the 5-in-a-row, whose rules are almost identical to the ancient Japanese Go-Moku. Along this line Harary asked if a player can achieve a translated copy of a given polymino P when the two players alternately take the squares of the board. Here we pose his question for general subsets of the board, and give a condition under which a draw is possible. Since a drawing strategy corresponds to a good 2-coloration of the underlying hypergraph, our result can be viewed as a derandomization of the Lovász Local Lemma.