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Claim. There is an integer $N$ such that every set of $N$ points in the plane in general position contains six points in convex position with no point of the set inside their convex hull. In the notation of Problem 216, $g(6)$ exists.

Covers. The existence half of the question at $k=6$: $g(6)$ exists. The proof gives no small bound on its value; Gerken proved the same theorem independently, with the bound $g(6)\le 1717$, 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: C. M. Nicolás, The empty hexagon theorem, Discrete & Computational Geometry 38 (2007), no. 2, 389–397. The site's remarks credit the existence of $g(6)$ to this paper and to Gerken's independently; the site's label, disproved, rests on Horton's result, so that remark is not acceptance of this partial claim.