Van der Waerden's theorem (1927): for any r colors and length k, there exists W(r,k) such that any r-coloring of {1,...,W(r,k)} contains a monochromatic k-term AP. Known: W(2,3)=9, W(2,4)=35, W(2,5)=178, W(2,6)=1132. Exact values are extremely hard to compute.
How to Use
Enter r (colors) and k (AP length):
- W(r,k): Exact or bounds
- Table: Known values
- Example: AP demonstration
The Theorem
No matter how you color {1,...,n} with r colors, if n ≥ W(r,k), some color class contains a k-term AP (a, a+d, a+2d, ..., a+(k-1)d). This is guaranteed regardless of the coloring strategy!
Growth Rate
W(r,k) grows incredibly fast — at least double-exponential in k. Gowers (1998) proved an upper bound, winning a Fields Medal partly for this work. The gap between known bounds is enormous.
Step-by-Step Instructions
- 1Enter colors r.
- 2Enter AP length k.
- 3Look up W(r,k).
- 4View bounds.
- 5Explore examples.