Van der Waerden's theorem

Van der Waerden's theorem is a theorem in the branch of mathematics called Ramsey theory. Van der Waerden's theorem states that for any given positive integers r and k, there is some number N such that if the integers {1, 2, ..., N} are colored, each with one of r different colors, then there are at least k integers in arithmetic progression whose elements are of the same color. The least such N is the Van der Waerden number W(r, k), named after the Dutch mathematician B. L. van der Waerden.[1]

This was conjectured by Pierre Joseph Henry Baudet in 1921. Waerden heard of it in 1926 and published his proof in 1927, titled Beweis einer Baudetschen Vermutung [Proof of Baudet's conjecture].[2][3][4]

  1. ^ van der Waerden, B. L. (1927). "Beweis einer Baudetschen Vermutung". Nieuw. Arch. Wisk. (in German). 15: 212–216.
  2. ^ L, van der WAERDEN B. (1927). "Beweis einer Baudetschen Vermutung". Nieuw Arch.Wiskunde. 15: 212–216.
  3. ^ Soifer, Alexander (2015), Soifer, Alexander (ed.), "Whose Conjecture Did Van der Waerden Prove?", The Scholar and the State: In Search of Van der Waerden, Basel: Springer, pp. 379–401, doi:10.1007/978-3-0348-0712-8_38, ISBN 978-3-0348-0712-8, retrieved 2024-01-17
  4. ^ L, van der WAERDEN B. (1971). "How the proof of Baudet conjecture was found". Studies in Pure Math. Reprinted in Chapter 33 of "The Mathematical Coloring Book".