Editing Van der Waerden's theorem (section)

==Example==
For example, when ''r'' = 2, you have two colors, say <span style="color:red;">red</span> and <span style="color:blue;">blue</span>. ''W''(2, 3) is bigger than 8, because you can color the integers from {1, ..., 8} like this:
{|class="wikitable"
|&nbsp;1&nbsp;
|&nbsp;2&nbsp;
|&nbsp;3&nbsp;
|&nbsp;4&nbsp;
|&nbsp;5&nbsp;
|&nbsp;6&nbsp;
|&nbsp;7&nbsp;
|&nbsp;8&nbsp;
|-
|&nbsp;'''<span style="color:blue;">B</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;
|&nbsp;'''<span style="color:blue;">B</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;
|}

and no three integers of the same color form an [[arithmetic progression]].  But you can't add a ninth integer to the end without creating such a progression.  If you add a <span style="color:red;">red 9</span>, then the <span style="color:red;">red 3</span>, <span style="color:red;">6</span>, and <span style="color:red;">9</span> are in arithmetic progression.  Alternatively, if you add a <span style="color:blue;">blue 9</span>, then the <span style="color:blue;">blue 1</span>, <span style="color:blue;">5</span>, and <span style="color:blue;">9</span> are in arithmetic progression.

In fact, there is no way of coloring 1 through 9 without creating such a progression (it can be proved by considering examples).  Therefore, ''W''(2, 3) is 9.