van der Waerden theorem

Target entity: van der Waerden theorem
Context triple: [Ramsey theory, hasCentralConcept, van der Waerden theorem]

• A. Szemerédi's theorem
  Szemerédi's theorem is a fundamental result in combinatorial number theory stating that any subset of the integers with positive upper density contains arbitrarily long arithmetic progressions.
• B. Graham–Rothschild theorem
  The Graham–Rothschild theorem is a fundamental result in Ramsey theory that generalizes classical partition theorems to higher-dimensional combinatorial structures.
• C. Steinhaus theorem
  The Steinhaus theorem is a fundamental result in measure theory stating that the difference set of any subset of the real numbers with positive Lebesgue measure contains an open interval around zero.
• D. Roth theorem
  Roth's theorem is a fundamental result in Diophantine approximation that gives an essentially optimal bound on how well algebraic irrational numbers can be approximated by rational numbers.
• E. Ramsey theory
  Ramsey theory is a branch of combinatorics that studies the conditions under which order or structure must appear within sufficiently large or complex mathematical objects.
• F. None of above. chosen
• G. Unsure - the case is ambiguous/there is not enough information to decide.

Target entity: van der Waerden theorem
Target entity description: The van der Waerden theorem is a fundamental result in Ramsey theory stating that any finite coloring of the positive integers contains arbitrarily long monochromatic arithmetic progressions.

• A. Szemerédi's theorem
  Szemerédi's theorem is a fundamental result in combinatorial number theory stating that any subset of the integers with positive upper density contains arbitrarily long arithmetic progressions.
• B. Graham–Rothschild theorem
  The Graham–Rothschild theorem is a fundamental result in Ramsey theory that generalizes classical partition theorems to higher-dimensional combinatorial structures.
• C. Steinhaus theorem
  The Steinhaus theorem is a fundamental result in measure theory stating that the difference set of any subset of the real numbers with positive Lebesgue measure contains an open interval around zero.
• D. Roth theorem
  Roth's theorem is a fundamental result in Diophantine approximation that gives an essentially optimal bound on how well algebraic irrational numbers can be approximated by rational numbers.
• E. Ramsey theory
  Ramsey theory is a branch of combinatorics that studies the conditions under which order or structure must appear within sufficiently large or complex mathematical objects.
• F. None of above. chosen