Editing Van der Waerden's theorem (section)

=== Proof in the case of ''W''(2, 3) ===
{| class="wikitable floatright" style="text-align:right
|+ ''W''(2, 3) table
! ''b'' !! colspan="5" | ''c''(''n''): color of integers
|-
! rowspan="2" | 0
| 1 || 2 || 3 || 4 || 5
|-
| &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:red;">R</span>'''&nbsp; || &nbsp;'''<span style="color:blue;">B</span>'''&nbsp;
|-
! rowspan="2" | 1
| 6 || 7 || 8 || 9 || 10
|-
| &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:red;">R</span>'''&nbsp;
|-
! …
| colspan="5" | …
|-
! rowspan="2" | 64
| 321 || 322 || 323 || 324 || 325
|-
| &nbsp;'''<span style="color:red;">R</span>'''&nbsp; || &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:red;">R</span>'''&nbsp;
|}
We will prove the special case mentioned above, that ''W''(2, 3) ≤ 325. Let ''c''(''n'') be a coloring of the integers {1, ..., 325}.  We will find three elements of {1, ..., 325} in arithmetic progression that are the same color.

Divide {1, ..., 325} into the 65 blocks {1, ..., 5}, {6, ..., 10}, ... {321, ..., 325}, thus each block is of the form {5''b'' + 1, ..., 5''b'' + 5} for some ''b'' in {0, ..., 64}. Since each integer is colored either <span style="color:red;">red</span> or <span style="color:blue;">blue</span>, each block is colored in one of 32 different ways.  By the [[pigeonhole principle]], there are two blocks among the first 33 blocks that are colored identically. That is, there are two integers ''b''<sub>1</sub> and ''b''<sub>2</sub>, both in {0,...,32}, such that

: ''c''(5''b''<sub>1</sub> + ''k'') = ''c''(5''b''<sub>2</sub> + ''k'')

for all ''k'' in {1, ..., 5}.  Among the three integers 5''b''<sub>1</sub> + 1, 5''b''<sub>1</sub> + 2, 5''b''<sub>1</sub> + 3, there must be at least two that are of the same color. (The [[pigeonhole principle]] again.)  Call these 5''b''<sub>1</sub> + ''a''<sub>1</sub> and 5''b''<sub>1</sub> + ''a''<sub>2</sub>, where the ''a''<sub>''i''</sub> are in {1,2,3} and ''a''<sub>1</sub> &lt; ''a''<sub>2</sub>.  Suppose (without loss of generality) that these two integers are both <span style="color:red;">red</span>.  (If they are both <span style="color:blue;">blue</span>, just exchange '<span style="color:red;">red</span>' and '<span style="color:blue;">blue</span>' in what follows.)

Let ''a''<sub>3</sub> = 2''a''<sub>2</sub>&nbsp;&minus;&nbsp;''a''<sub>1</sub>. If 5''b''<sub>1</sub> + ''a''<sub>3</sub> is <span style="color:red;">red</span>, then we have found our arithmetic progression: 5''b''<sub>1</sub>&nbsp;+&nbsp;''a''<sub>''i''</sub> are all <span style="color:red;">red</span>.	

Otherwise, 5''b''<sub>1</sub> + ''a''<sub>3</sub> is <span style="color:blue;">blue</span>. Since ''a''<sub>3</sub> ≤ 5,  5''b''<sub>1</sub> + ''a''<sub>3</sub> is in the ''b''<sub>1</sub> block, and since the ''b''<sub>2</sub> block is colored identically, 5''b''<sub>2</sub> + ''a''<sub>3</sub> is also <span style="color:blue;">blue</span>.

Now let ''b''<sub>3</sub> = 2''b''<sub>2</sub>&nbsp;&minus;&nbsp;''b''<sub>1</sub>. Then ''b''<sub>3</sub> ≤ 64. Consider the integer  5''b''<sub>3</sub> + ''a''<sub>3</sub>, which must be ≤ 325. What color is it?

If it is <span style="color:red;">red</span>, then 5''b''<sub>1</sub> + ''a''<sub>1</sub>, 5''b''<sub>2</sub> + ''a''<sub>2</sub>, and 5''b''<sub>3</sub> + ''a''<sub>3</sub> form a <span style="color:red;">red</span> arithmetic progression. But if it is <span style="color:blue;">blue</span>, then 5''b''<sub>1</sub> + ''a''<sub>3</sub>, 5''b''<sub>2</sub> + ''a''<sub>3</sub>, and 5''b''<sub>3</sub> + ''a''<sub>3</sub> form a <span style="color:blue;">blue</span> arithmetic progression. Either way, we are done.