(p,q)-theorem

Target entity: (p,q)-theorem
Context triple: [Helly’s theorem, relatedTo, (p,q)-theorem]

• A. Radon’s theorem
  Radon’s theorem is a fundamental result in convex geometry stating that any set of sufficiently many points in Euclidean space can be partitioned into two disjoint subsets whose convex hulls intersect.
• B. Helly’s theorem
  Helly’s theorem is a fundamental result in convex geometry that gives conditions under which a family of convex sets in Euclidean space has a nonempty common intersection.
• C. Sylvester–Gallai theorem
  The Sylvester–Gallai theorem is a result in incidence geometry stating that for any finite set of points in the Euclidean plane not all on a single line, there exists a line that passes through exactly two of the points.
• D. Carathéodory’s theorem in convex geometry
  Carathéodory’s theorem in convex geometry is a fundamental result stating that any point in the convex hull of a set in ℝⁿ can be expressed as a convex combination of at most n+1 points from that set.
• E. Erdős–Szekeres theorem
  The Erdős–Szekeres theorem is a fundamental result in combinatorial geometry that guarantees the existence of large convex polygons within sufficiently large sets of points in the plane in general position.
• F. None of above. chosen
• G. Unsure - the case is ambiguous/there is not enough information to decide.

Target entity: (p,q)-theorem
Target entity description: The (p,q)-theorem is a fundamental result in combinatorial geometry that generalizes Helly-type intersection properties by guaranteeing large intersecting subfamilies within set systems under certain local intersection conditions.

• A. Radon’s theorem
  Radon’s theorem is a fundamental result in convex geometry stating that any set of sufficiently many points in Euclidean space can be partitioned into two disjoint subsets whose convex hulls intersect.
• B. Helly’s theorem
  Helly’s theorem is a fundamental result in convex geometry that gives conditions under which a family of convex sets in Euclidean space has a nonempty common intersection.
• C. Sylvester–Gallai theorem
  The Sylvester–Gallai theorem is a result in incidence geometry stating that for any finite set of points in the Euclidean plane not all on a single line, there exists a line that passes through exactly two of the points.
• D. Carathéodory’s theorem in convex geometry
  Carathéodory’s theorem in convex geometry is a fundamental result stating that any point in the convex hull of a set in ℝⁿ can be expressed as a convex combination of at most n+1 points from that set.
• E. Erdős–Szekeres theorem
  The Erdős–Szekeres theorem is a fundamental result in combinatorial geometry that guarantees the existence of large convex polygons within sufficiently large sets of points in the plane in general position.
• F. None of above. chosen