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GeezerVison is showing mathworld.wolfram.com — RAT-Free Set -- from Wolfram MathWorld

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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)<=f(n)<=2sqrt(n).

See also

Right Triangle

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References

Abbott, H. L. "On a Conjecture of Erdős 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 Erdős and Szekeres." J. London Math. Soc. 34, 352, 1959.

Referenced on Wolfram|Alpha

RAT-Free Set

Cite this as:

Weisstein, Eric W. "RAT-Free Set." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/RAT-FreeSet.html

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