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Statement

Setting (p. 52). $f(n)$ is the largest integer such that any set of $n$ points in the plane, no three on a line, contains at least $f(n)$ convex subsets.

Conjecture (p. 53). Right after proving inequality (2), $n^{c_1\log n}<f(n)<n^{c_2\log n}$, the paper says that there is probably a constant $c$ with

$$\lim_{n\to\infty}\frac{\log f(n)}{(\log n)^2}=c.$$

The print writes the limit with $n=\infty$ beneath it. The paper offers no argument for the guess.

Source. P. Erdős, Some more problems on elementary geometry, Austral. Math. Soc. Gaz. 5 (1978), no. 2, 52--54: the definition of $f(n)$ on p. 52 and the conjecture on p. 53. The edition read is identified on the source card.

Read depth. Claims checked: the definition and the displayed limit were read on the page images of pp. 52--53.

Proof pointer

None; the statement is a conjecture.

Dependencies

Inequality (2) shows that the ratio lies between $c_1$ and $c_2$, which is the context of the guess.

Bears on

• Problem 838: the problem's question whether this limit exists is the paper's conjecture, posed there as a question; the paper does not settle it.