Mathematics Colloquia and Seminars

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According to a celebrated theorem of Sylvester and Gallai, any finite set $S$ of non-collinear points in the plane has two elements whose connecting line does not pass through any other point in $S$. Erd\H os noticed that this result immediately implies that any set of $n$ non-collinear points in the plane determines at least $n$ different connecting lines. Equality is attained if and only if all but one of the points are on a line. In the same spirit, Scott posed two similar questions in 1970: (1) Is it true that the number of different directions assumed by the connecting lines of $n>3$ non-collinear points in the plane is at least $n-1$? (2) Is it true that the number of different directions assumed by the connecting lines of $n>5$ non-coplanar points in $3$-space is at least $2n-3$? The first question was answered in the affirmative by Ungar in 1982, using allowable sequences (see {\em Proofs from the Book} by Aigner and Ziegler). We outline a completely elementary argument of Pinchasi, Sharir, and the speaker that solves the second problem of Scott. We also mention several open problems.

There will be a special lunch of the Undergraduate Math Majors with Prof. Pach.