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Claim. For every $k$ there is a set of $2^k$ points in the plane that contains no empty convex heptagon: no seven of its points are the vertices of a convex heptagon whose interior is free of the set. Hence $g(7)$ does not exist, and neither does $g(k)$ for any $k\ge7$, since an empty convex $k$-gon contains an empty convex heptagon among its vertices. The problem asks whether $g(k)$ exists for every $k$; Horton's construction answers no. The problem's wording and the site's page omit the general-position convention; Horton defines $g(n)$ over sets with no three collinear, as the literature on $g(k)$ does, and $S_k$ is such a set.

The construction. Horton's set $S_k$ consists of the points $(i,d(i))$ for $0\le i<2^k$, where $d(i)$ is read off the binary digits of $i$ with a base $c=2^k+1$. Splitting on the lowest binary digit cuts $S_k$ into a bottom half $B$ and a top half $T$ that are scaled translates of each other, and every point of $T$ lies above every line through two points of $B$. An empty convex polygon lying in one half maps affinely onto an empty convex polygon of the smaller set, so one may assume it meets both halves. Such a polygon has at most three vertices in $B$ and three in $T$, hence at most six. The source card records the paper's observations and the counting step.

What remains of the question. The function exists for small $k$: $g(4)=5$ (Erdős), $g(5)=10$ (Harborth), and $g(6)$ exists by the independent proofs of Nicolás and Gerken, with $g(6)=30$ established by Heule and Scheucher. Horton's note already records $g(5)=10$ and leaves the hexagon case open.

Acceptance. The paper is refereed: J. D. Horton, Sets with no empty convex 7-gons, Canad. Math. Bull. 26 (1983), no. 4, 482–484. The curator of erdosproblems.com, Thomas Bloom, labels the problem disproved and credits Horton's paper for the nonexistence of $g(k)$ for $k\ge7$.