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Updated Oct 8

Problem 504

claims/: The 2 claim pages of Problem 504, one per claimant's result; the problem's standing derives from them.

Statement. Let $\alpha_n$ be the supremum of all $0\leq \alpha\leq \pi$ such that in every set $A\subset \mathbb{R}^2$ of size $n$ there exist three distinct points $x,y,z\in A$ such that the angle determined by $xyz$ is at least $\alpha$ . Determine $\alpha_n$ .

Status. Solved: the site labels the problem SOLVED, crediting Sendov's 1993 determination of $\alpha_N$ for every $N\ge4$ ( $\alpha_3=\pi/3$ is the equilateral triangle), recorded on the Sendov claim page; the earlier values at powers of two are on the Erdős–Szekeres claim page.

Source. erdosproblems.com/504, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #504, https://www.erdosproblems.com/504.

References.

[ErSz60] Erdős, P. and Szekeres, G., On some extremum problems in elementary geometry. Ann. Univ. Sci. Budapest. Eötvös Sect. Math. (1960/61), 53-62.
[Se92] Sendov, Bl., On a conjecture of P. Erdős and G. Szekeres. C. R. Acad. Bulgare Sci. (1992), 17-20.
[Se93] Sendov, Bl., Angles in a plane configuration of points. C. R. Acad. Bulgare Sci. (1993), 27-30.
[Sz41] Szekeres, Gy., On an extremum problem in the plane. Amer. J. Math. (1941), 208-210.

Formalization. None recorded.

Progress

Not yet compiled.

Known Results

Not yet compiled.

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 (9)

problems/discrete_geometry/E0504/claims Discrete and Convex Geometry discrete_geometry/erdos_1960_extremum_problems_elementary_geometry Theorem 1 (p. 54; Theorem 1*, p. 59): every plane configuration of 2^n points, n >= 3, has an angle greater than (1 - 1/n)pi Theorem 2 (p. 60): every plane configuration of N = 2^n - k points, 0 < k < 2^{n-1}, has an angle >= (1 - 1/n - k/2N)pi Theorem 3 (p. 61, note added in proof): every plane configuration of 2^n - 1 points has an angle not less than (1 - 1/n)pi Small values, p. 54: alpha(3) = pi/3, alpha(4) = pi/2, alpha(5) = 3pi/5, alpha(6) = alpha(7) = alpha(8) = 2pi/3 discrete_geometry/erdos_1981_applications_graph_theory_combinatorial_methods_number Display (2), p. 139: convex k-gons with all but two angles greater than π − ε, and the angle results behind t_ε

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