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An n-in-a-row type game

Abstract

We consider a Maker-Breaker type game on the plane, in which each player takes tt points on their ttht^\textrm{th} turn. Maker wins if he obtains nn points on a line (in any direction) without any of Breaker's points between them. We show that, despite Maker's apparent advantage, Breaker can prevent Maker from winning until about his nthn^\textrm{th} turn. We actually prove a stronger result: that Breaker only needs to play ω(logt)\omega(\log t) points on his ttht^\textrm{th} turn to prevent Maker from winning until this time. We also consider the situation when the number of points claimed by Maker grows at other speeds, in particular, when Maker claims tαt^\alpha points on his ttht^\textrm{th} turn.Comment: 17 page

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