Sylvester–Gallai theorem

Every finite set of noncollinear points in the Euclidean plane determines a line containing exactly two of the points. Such a line is called an ordinary line.

Ordinary line: A line containing exactly two points from a specified finite point set. The theorem guarantees at least one such line in every noncollinear finite point set.
James Joseph Sylvester: A British mathematician whose work ranged across algebra, number theory, and geometry. He posed the ordinary-line question in 1893.
Collinearity: The property of points lying on a single straight line. The theorem assumes the whole point set is not collinear.
Szemerédi–Trotter theorem: A sharp upper bound on incidences between a finite set of points and a finite set of lines in the plane. Both results turn geometric incidence structure into constraints on finite configurations.
Finite-field Sylvester–Gallai theorem: A family of results examining ordinary-line analogues for point sets over finite fields. Finite fields can admit noncollinear sets with no line containing exactly two selected points.
Proof of the Sylvester–Gallai theorem: A mathematical demonstration that every finite noncollinear planar point set determines an ordinary line. Its minimal-distance argument shows how the theorem forces a pair-only line.
Paul Erdős: A Hungarian mathematician known for prolific collaborations and contributions to combinatorics and number theory. He published a short proof of the theorem in 1943.
Finite set: A set containing a limited number of distinct elements. Finiteness is essential to the theorem's guarantee.
Arrangement of lines: The subdivision of a plane by a finite collection of lines and their intersections. Dualizing points to lines converts ordinary-line questions into properties of line arrangements.
Hesse configuration: A configuration of nine points and twelve lines in the complex projective plane, with three points on each line. It shows why the theorem's restriction to real Euclidean geometry matters.

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Big-line-big-clique conjectureCompared with: It guarantees only a two-point line, far weaker than the conjecture’s large-subset alternative.