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Updated Oct 7

Problem 838

Statement. Let $f(n)$ be maximal such that any $n$ points in $\mathbb{R}^2$ , with no three on a line, determine at least $f(n)$ different convex subsets. Estimate $f(n)$ - in particular, does there exist a constant $c$ such that

\lim \frac{\log f(n)}{(\log n)^2}=c?

Status. Open.

Source. erdosproblems.com/838, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #838, https://www.erdosproblems.com/838.

References.

[Er78c] Erdős, P., Some more problems on elementary geometry. Austral. Math. Soc. Gaz. (1978), 52-54.

Formalization. None recorded.

Progress

Not yet compiled.

Known Results

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Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

Linked from (8)

Discrete and Convex Geometry discrete_geometry/erdos_1978_more_problems_elementary_geometry Conjecture (p. 53): log f(n)/(log n)^2 probably tends to a constant Inequality 2 (p. 53): n^{c_1 log n} < f(n) < n^{c_2 log n} for the least number of convex subsets discrete_geometry/erdos_1981_applications_graph_theory_combinatorial_methods_number Display (4), p. 140: n points with no three on a line have between n^(c₁ log n) and n^((1+o(1)) log n / log 2) convex subsets at the minimum On the Erdős-Szekeres convex polygon problem Theorem 1.1 (p. 1): ES(n) <= 2^{n+6n^{2/3} log n} for all n >= n_0

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