## On Mutually Avoiding Sets


Pavel Valtr
Department of Applied Mathematics
Charles University
Czech Republic
email: valtr@CSPGUK11.bitnet
Report B 94-05
January 94

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Two finite sets of points in the plane are called mutually avoiding if any straight line passing through two points of anyone of these two sets does not intersect the convex hull of the other set. For any integer $n,$ we construct a set of $n$ points in general position in the plane which contains no pair of mutually avoiding sets of size more than $\bigO(\sqrt n).$ The given bound is tight up to a constant factor, since Aronov {\it et~al.}~\cite{Aronov} showed a polynomial-time algorithm for finding two mutually avoiding sets of size $\Omega(\sqrt n)$ in any set of $n$ points in general position in the plane.