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Claim. Every set of points in the plane in general position that contains nine points in convex position contains six points in convex position with no point of the set inside their convex hull. Hence $g(6)$ exists in the notation of Problem 216, and it is at most the number of points forcing a convex 9-gon, which is at most 1717 by Tóth and Valtr's bound.

Covers. The existence half of the question at $k=6$, with the bound $g(6)\le 1717$. Nicolás proved the existence independently, and Heule and Scheucher later determined $g(6)=30$. Nothing is claimed about $k\ge7$, where $g(k)$ does not exist by Horton's result.

Acceptance. The paper is refereed: T. Gerken, Empty convex hexagons in planar point sets, Discrete & Computational Geometry 39 (2008), no. 1–3, 239–272, published online 2007-09-11. The site's remarks credit the existence of $g(6)$ to this paper and to Nicolás's independently; the site's label, disproved, rests on Horton's result, so that remark is not acceptance of this partial claim.