Claims

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1978_01_01_harborth: Every set of 10 points in general position in the plane contains an empty convex pentagon, and some 9 points contain none, so $g(5)=10$.

1983_12_01_horton: Horton constructs arbitrarily large planar point sets in general position with no empty convex heptagon, so $g(k)$ does not exist for $k\ge7$ and the question whether $g(k)$ always exists is answered negatively.

2007_09_01_nicolas: Every sufficiently large set of points in general position in the plane contains an empty convex hexagon, so $g(6)$ exists.

2007_09_11_gerken: Every set of points in general position in the plane that contains a convex 9-gon contains an empty convex hexagon, so $g(6)$ exists and is at most 1717.

2024_03_01_heule_scheucher: A satisfiability proof that every set of 30 points in general position in the plane contains an empty convex hexagon; with Overmars's 29-point set this gives $g(6)=30$, the last value of $g(k)$ that exists.

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