On a generalization of Kaplansky's game

In this paper, games of the following general kind are studied: Two players move alternately by selecting unselected integer coordinate points in the plane. On each move, the first player selects exactly r points and the second player selects exactly one point. The first player wins if he can select p points on a line having none of his opponent's points before his opponent selects q points on a line having none of his own. If this latter eventuality occurs first, the second player wins. It is shown that if p ? c(r)q, then the second player can always win.