Big-line-big-clique conjecture

The conjecture that every finite set of planar points contains either a large collinear subset or a large subset with no three collinear.

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Ordinary line: A line containing exactly two points of a given finite point set. The conjecture asks whether avoiding large collinear sets forces large subsets governed by such lines.
Paul Erdős: A Hungarian mathematician known for influential work in combinatorics, number theory, and discrete mathematics. Erdős posed the problem as part of his program on geometric configurations.
Sylvester–Gallai theorem: A theorem stating that every finite noncollinear real point set has a line containing exactly two of its points. It guarantees only a two-point line, far weaker than the conjecture’s large-subset alternative.
Incidence geometry: The study of incidences between discrete points, lines, and other geometric objects. Incidence bounds may constrain configurations that avoid both large collinear and large independent subsets.
General position: A configuration in which no unwanted degeneracy occurs, as specified by the problem. The no-three-collinear alternative is the planar point-set version of general position.
Erdős–Szekeres problem: A family of questions about the smallest point-set size forcing a convex polygon of a given size. It supplies a neighboring line of inquiry into unavoidable order among points.
Erdős–Szekeres theorem: A theorem guaranteeing a convex polygon of prescribed size in every sufficiently large planar point set in general position. It proves a different structural guarantee and assumes away collinear triples.
Szemerédi–Trotter theorem: A theorem giving an asymptotically sharp upper bound on incidences between points and lines in the plane. Its incidence estimates offer tools for controlling line-rich point configurations.
Ramsey theory: The study of conditions that force structured subconfigurations in sufficiently large systems. The conjecture is a geometric Ramsey-type demand for one of two contrasting structures.
Erdős–Hajnal conjecture: A conjecture that every hereditary graph class excluding a fixed induced pattern contains polynomial-size structured subsets. It expresses a related principle: forbidding one pattern can force a large structured subset.