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The threshold numbers proven to exist by van der Waerden's Theorem. The first few are 1, 3, 9, 35, 178, ... (Sloane's A005346).
References
Goodman, J. E. and O'Rourke, J. (Eds.). Handbook of Discrete & Computational Geometry.
Boca Raton, FL: CRC Press, p. 159, 1997.
Honsberger, R. More Mathematical Morsels. Washington, DC: Math. Assoc. Amer., p. 29, 1991.
Sloane, N. J. A. Sequence
A005346/M2819
in ``An On-Line Version of the Encyclopedia of Integer Sequences.''
http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S.
The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.