Editing Van der Waerden's theorem (section)

=== Proof in the case of ''W''(3, 3) ===
{| class="wikitable floatright" style="text-align:right
|+ W(3, 3) table<br />''g''=2·3<sup>7·(2·3<sup>7</sup>&nbsp;+&nbsp;1)</sup>&nbsp;,<br />''m''=7(2·3<sup>7</sup>&nbsp;+&nbsp;1)
! ''b'' !! colspan="5" | ''c''(''n''): color of integers
|-
! rowspan="2" | 0
| 1 || 2 || 3 || … || ''m''
|-
| &nbsp;'''<span style="color:limegreen;">G</span>'''&nbsp; || &nbsp;'''<span style="color:red;">R</span>'''&nbsp; || &nbsp;'''<span style="color:red;">R</span>'''&nbsp; || … || &nbsp;'''<span style="color:blue;">B</span>'''&nbsp;
|-
! rowspan="2" | 1
| ''m'' + 1 || ''m'' + 2 || ''m'' + 3 || … || 2''m''
|-
| &nbsp;'''<span style="color:blue;">B</span>'''&nbsp; || &nbsp;'''<span style="color:red;">R</span>'''&nbsp; || &nbsp;'''<span style="color:limegreen;">G</span>'''&nbsp; || … || &nbsp;'''<span style="color:red;">R</span>'''&nbsp;
|-
! …
| colspan="5" | …
|-
! rowspan="2" | ''g''
| ''gm'' + 1 || ''gm'' + 2 || ''gm'' + 3 || … || (''g'' + 1)''m''
|-
| &nbsp;'''<span style="color:blue;">B</span>'''&nbsp; || &nbsp;'''<span style="color:red;">R</span>'''&nbsp; || &nbsp;'''<span style="color:blue;">B</span>'''&nbsp; || … || &nbsp;'''<span style="color:limegreen;">G</span>'''&nbsp;
|}
A similar argument can be advanced to show that ''W''(3, 3) ≤ 7(2·3<sup>7</sup>+1)(2·3<sup>7·(2·3<sup>7</sup>+1)</sup>+1). One begins by dividing the integers into  2·3<sup>7·(2·3<sup>7</sup>&nbsp;+&nbsp;1)</sup>&nbsp;+&nbsp;1 groups of 7(2·3<sup>7</sup>&nbsp;+&nbsp;1) integers each; of the first 3<sup>7·(2·3<sup>7</sup>&nbsp;+&nbsp;1)</sup>&nbsp;+&nbsp;1 groups, two must be colored identically.

Divide each of these two groups into 2·3<sup>7</sup>+1 subgroups of 7 integers each; of the first 3<sup>7</sup>&nbsp;+&nbsp;1 subgroups in each group, two of the subgroups must be colored identically.  Within each of these identical subgroups, two of the first four integers must be the same color, say <span style="color:red;">red</span>; this implies either a <span style="color:red;">red</span> progression or an element of a different color, say <span style="color:blue;">blue</span>, in the same subgroup.

Since we have two identically-colored subgroups, there is a third subgroup, still in the same group that contains an element which, if either <span style="color:red;">red</span> or <span style="color:blue;">blue</span>, would complete a <span style="color:red;">red</span> or <span style="color:blue;">blue</span> progression, by a construction analogous to the one for ''W''(2, 3). Suppose that this element is <span style="color:limegreen;">green</span>. Since there is a group that is colored identically, it must contain copies of the <span style="color:red;">red</span>, <span style="color:blue;">blue</span>, and <span style="color:limegreen;">green</span> elements we have identified; we can now find a pair of <span style="color:red;">red</span> elements, a pair of <span style="color:blue;">blue</span> elements, and a pair of <span style="color:limegreen;">green</span> elements that 'focus' on the same integer, so that whatever color it is, it must complete a progression.