RAT-Free Set

A RAT-free (``right angle triangle-free'') set is a set of points, no three of which determine a Right Triangle. Let $f(n)$ be the largest integer such that a RAT-free subset of size $f(n)$ is guaranteed to be contained in any set of $n$ coplanar points. Then the function $f(n)$ is bounded by

$$\sqrt{n}\leq f(n)\leq 2\sqrt{n}.$$

References

Abbott, H. L. ``On a Conjecture of Erdös and Silverman in Combinatorial Geometry.'' J. Combin. Th. A 29, 380-381, 1980.

Chan, W. K. ``On the Largest RAT-FREE Subset of a Finite Set of Points.'' Pi Mu Epsilon 8, 357-367, 1987.

Honsberger, R. More Mathematical Morsels. Washington, DC: Math. Assoc. Amer., pp. 250-251, 1991.

Seidenberg, A. ``A Simple Proof of a Theorem of Erdös and Szekeres.'' J. London Math. Soc. 34, 352, 1959.