In discrete geometry, Beck's theorem refers to any of several related results proved by Jozsef Beck in a 1983 paper published in Combinatorica, establishing lower bounds on the number of lines determined by a finite set of points in the plane, where a line counts as determined if it passes through at least two of the points. The theorem's original proof used a constant later shown to be far from optimal, and the true best possible constants remain unknown. A related and stronger result, the Erdos-Beck theorem, implies Beck's theorem though it was not stated in Beck's own paper.
Facts
StatementFinite collections of points in the plane fall into one of two extremes: a large fraction of points lie on a single line, or a large number of lines are needed to connect all the points. 1 Classification
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Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
In Branch
Source Beck's theorem (geometry) - Wikipedia
Sources
1. Beck's theorem (geometry) - Wikipedia
Beck's theorem
Beck's theorem says that finite collections of points in the plane fall into one of two extremes
In Branch: Discrete Geometry, Lead sentence
In discrete geometry, Beck's theorem is any of several different results, two of which are given below.
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