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Source. The paragraphs spanning pp. 51--52 of P. Erdős, Combinatorial problems in geometry, Math. Chronicle 12 (1983), 35--54, the transcript of an invited address at the 17th New Zealand Mathematics Colloquium (Dunedin, 17--19 May 1982), as named on the source card. The lecture numbers none of its statements; the pages are the journal's own.

Statement

Definition (p. 51). $n(k)$, if it exists, is the smallest integer such that any $n(k)$ points in the plane, no three on a line, contain a convex $k$-gon containing none of the points in its interior.

Reported values (pp. 51--52).

1. $n(4)=5$, which the lecture calls trivial and sketches.
2. $n(5)=10$, proved by Harborth.
3. Nobody had proved that $n(6)$ exists. Its non-existence would mean that for every $t$ there are $t$ points in the plane, no three on a line, with every convex hexagon containing one of the points in its interior. Harborth suggested that $n(6)$ may exist while $n(7)$ does not; Erdős says he does not know.

Erdős says he noticed the variant in 1976.

Read depth. Claims checked: the passage was read clause by clause on the page images of the print. A second reader checked the statement, hypotheses, label and page against the print.

Proof pointer

The paper sketches only $n(4)=5$; it cites Harborth for $n(5)=10$.

Dependencies

problem_p49 for the convex polygon problem it varies.

Bears on

• Problem 216: the lecture's $n(k)$ is the problem's $g(k)$ with the no-three-on-a-line condition stated explicitly. It reports $n(4)=5$ and $n(5)=10$ and leaves the existence of $n(6)$ and $n(7)$ open; it proves nothing toward the general question.