A RAT-free ("right angle triangle-free") set is a set of points, no three of which determine a right triangle. Let be the largest integer such that a RAT-free subset of size
is guaranteed to be contained in
any set of
coplanar points. Then the function is bounded by
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R. More
Mathematical Morsels. Washington, DC: Math. Assoc. Amer., pp. 250-251,
1991.Seidenberg, A. "A Simple Proof of a Theorem of Erdős
and Szekeres." J. London Math. Soc.34, 352, 1959.
be the largest integer such that a RAT-free subset of size
is guaranteed to be contained in
any set of 
coplanar points. Then the function 
is bounded by
