van der Waerden's Theorem

• Given a bit string long enough, there will always be a sequence of same-valued bits whose positions are in arithmetic sequence
• Holds for many colors

Example: k=4
0010001001110010000                 
 ^ ^ ^ ^       ^^^^ 0) (1,2), (15,1)
   ^    ^    ^    ^    (3,5)        
  ^   ^   ^   ^     1) (2,4)