Editing Van der Waerden's theorem (section)

{{Short description|Theorem in Ramsey theory}}
'''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 [[integer]]s ''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'',&nbsp;''k''), named after the Dutch mathematician [[Bartel Leendert van der Waerden|B. L. van der Waerden]].<ref>{{cite journal |author-link=Bartel Leendert van der Waerden |first=B. L. |last=van der Waerden |title=Beweis einer Baudetschen Vermutung|language=de |journal=Nieuw. Arch. Wisk. |volume=15 |year=1927 |pages=212–216 }}</ref>

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]''.<ref>{{Cite journal |last=L |first=van der WAERDEN B. |date=1927 |title=Beweis einer Baudetschen Vermutung |url=https://cir.nii.ac.jp/crid/1573387449045250304 |journal=Nieuw Arch.Wiskunde |volume=15 |pages=212–216}}</ref><ref>{{Citation |last=Soifer |first=Alexander |title=Whose Conjecture Did Van der Waerden Prove? |date=2015 |url=https://doi.org/10.1007/978-3-0348-0712-8_38 |work=The Scholar and the State: In Search of Van der Waerden |pages=379–401 |editor-last=Soifer |editor-first=Alexander |access-date=2024-01-17 |place=Basel |publisher=Springer |language=en |doi=10.1007/978-3-0348-0712-8_38 |isbn=978-3-0348-0712-8|url-access=subscription }}</ref><ref>{{Cite journal |last=L |first=van der WAERDEN B. |date=1971 |title=How the proof of Baudet conjecture was found |url=https://cir.nii.ac.jp/crid/1574231873975387904 |journal=Studies in Pure Math. |at=Reprinted in Chapter 33 of "The Mathematical Coloring Book"}}</ref>