Editing Schur's theorem (section)

== Ramsey theory ==
{{wikibooks|Combinatorics|Schur's Theorem|Proof of Schur's theorem}}
In [[Ramsey theory]], '''Schur's theorem''' states that for any [[Partition of a set|partition]] of the [[positive integer]]s into a finite number of parts, one of the parts contains three integers ''x'', ''y'', ''z'' with 

:<math>x + y = z.</math>

For every positive integer ''c'', ''S''(''c'') denotes the smallest number ''S'' such that for every partition of the integers <math>\{1,\ldots, S \}</math> into ''c'' parts, one of the parts contains integers ''x'', ''y'', and ''z'' with <math>x + y = z</math>. Schur's theorem ensures that ''S''(''c'') is well-defined for every positive integer ''c''. The numbers of the form ''S''(''c'') are called '''Schur's number'''s.

[[Folkman's theorem]] generalizes Schur's theorem by stating that there exist arbitrarily large sets of integers, all of whose [[empty sum|nonempty]] sums belong to the same part.

Using this definition, the only known Schur numbers are ''S''(n) {{=}} 2, 5, 14, 45, and 161 ({{oeis|A030126}}) The [[mathematical proof|proof]] that {{nobr|''S''(5) {{=}} 161}} was announced in 2017 and required 2 [[petabyte]]s of space.<ref>{{cite arXiv |last=Heule |first=Marijn J. H. |author-link=Marijn Heule|title=Schur Number Five |eprint=1711.08076 |date=2017 |class=cs.LO }}</ref><ref>{{Cite web|title=Schur Number Five|url=https://www.cs.utexas.edu/~marijn/Schur/|access-date=2021-10-06|website=www.cs.utexas.edu}}</ref>