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Claim. For every $n\ge3$ there is a set of $2^{n-2}$ points in the plane, no three on a line, containing no $n$ points in convex position. This is the construction of Section 2 of P. Erdős and G. Szekeres, On some extremum problems in elementary geometry, Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 3--4 (1960/61), 53--62, built in Cartesian coordinates from convex and concave sequences of points. In the notation of Problem 107 it gives

$$f(n)\ge2^{n-2}+1\qquad(n\ge3),$$

which with the authors' earlier upper bound $f(n)\le\binom{2n-4}{n-2}+1$ brackets $f(n)$; the paper conjectures that the lower bound is the truth and notes that this is known for $n\le5$. Library home erdos_1960_extremum_problems_elementary_geometry.

Covers. The lower bound $f(n)\ge2^{n-2}+1$ for every $n\ge3$, one of the two inequalities of the conjectured equality $f(n)=2^{n-2}+1$. Not covered: the upper bound $f(n)\le2^{n-2}+1$, open for every $n\ge7$; it holds at $n=4$ by Klein's proposition, at $n=5$ by the result the site credits to Makai and Turán, and at $n=6$ by Szekeres and Peters.

Depends on. Nothing in this wiki; the construction is self-contained.

Acceptance. Refereed: the paper appeared in the Annales Universitatis Scientiarum Budapestinensis de Rolando Eötvös Nominatae, Sectio Mathematica, a journal. The site's commentary credits the lower bound to this paper, but the site labels the problem FALSIFIABLE, which settles nothing, so that credit is not listed as reviewed.

Dating. The journal volume carries the years 1960/61 and no month; the page is dated by the earlier year, and the day and month are placeholders.