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The STW Theorem.

For any positive integer N, any sufficiently large finite set of points in the plane in general position has a subset of N points that form the vertices of a convex polygon.
The proof appeared in the same paper that proves the Erd's–Szekeres theorem on monotonic subsequences in sequences of numbers.
Denoting by f(N) the minimal possible M for a set of M points in general position must contain a convex N-gon, it is known that
f(3) = 3, trivially.
f(4) = 5.[2]
f(5) = 9.[3] A set of eight points with no convex pentagon is shown in the illustration, demonstrating that f(5) > 8; the more difficult part of the proof is to show that every set of nine points in general position contains the vertices of a convex pentagon.
f(6) = 17.[4]
The value of f(N) is unknown for all N > 6; by the result of Erd's & Szekeres (1935) it is known to be finite.
On the basis of the known values of f(N) for N = 3, 4 and 5, Erd's and Szekeres conjectured in their original paper that

They proved later, by constructing explicit examples, that
[5]
but the best known upper bound when N ? 7 is
[6]