Richard Voepel, Rutgers University

Abstract

First introduced in 1917 by Henry Dudeney, the No-Three-In-Line problem asks for the maximum number of points that can be placed in an N by N grid such that no three are collinear. While there have been results concerning lower bounds for this number, non-trivial upper bounds remain largely conjectural. But this is not the only problem of this form to receive attention; one may consider generalizations to higher dimensions, asking for no three points to be collinear in an N by N by N grid, or for no four points to be coplanar. We present select results for these problems, and propose further cases to study.