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Claim. Among any five points in the plane, no three on a line, some four are the vertices of a convex quadrilateral. This is the proposition of Esther Klein with which P. Erdős and G. Szekeres, A combinatorial problem in geometry, Compositio Math. 2 (1935), 463--470, open their paper, with her proof: if the convex hull of the five points has four or five vertices, four of them form a convex quadrilateral; if it is a triangle, the line through the two interior points leaves two of the triangle's vertices on one side, and those two with the two interior points form a convex quadrilateral. Four points, three vertices of a triangle and one interior point, contain no convex quadrilateral, so in the notation of Problem 107

$$f(4)=5=2^{2}+1,$$

the value the paper records (as $N_0(4)=5$) beside $N_0(3)=3$ and, attributed to E. Makai without a proof, $N_0(5)=9$. The paper's general theorem, that $f(n)$ is finite for every $n$ with $f(n)\le\binom{2n-4}{n-2}+1$, exceeds $2^{n-2}+1$ for every $n\ge5$ and settles no further instance. Library home erdos_1935_combinatorial_problem_geometry.

Covers. The instance $n=4$ of the conjectured equality $f(n)=2^{n-2}+1$. Not covered: every $n\ge5$; the instance $n=6$ is Szekeres and Peters's, and the lower bound for every $n$ is the Erdős--Szekeres construction.

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

Acceptance. Refereed: the paper appeared in Compositio Mathematica, a journal. The site's commentary records $f(4)=5$ and credits it to Klein, but the site labels the problem FALSIFIABLE, which settles nothing, so that record is not listed as reviewed. The claimants are the authors who published the result; Klein's authorship of the proposition is the paper's own attribution.

Dating. The journal volume carries the year 1935 and no month; the day and month are placeholders.