15
Van der Waerden's theorem says that
For any given positive integers
r
andk
, there is some numberN
such that if the integers{1, 2, ..., N}
are colored, each with one ofr
different colors, then there are at leastk
integers in arithmetic progression all of the same color. The least suchN
is the Van der Waerden numberW(r, k)
.
Your goal is to compute the Van der Waerden number W(r, k)
given positive-integer inputs r
and k
. Fewest bytes wins.
Beware that this function grows extremely quickly and is time-consuming to compute. Even W(4, 4)
is unknown. You may assume your code runs on an ideal computer with unlimited resources (time, memory, stack depth, etc). Your code must theoretically give the right answer even for values for which the answer is not known.
Built-ins that compute this function are not allowed.
Example
For r = 2
colors and progressions of length k = 3
, there is a length-8
sequence that avoids such a progression, i.e. 3
equally-spaced elements of the same color:
B R R B B R R B
But, there's no such length-9
sequence, so W(2, 3) == 9
. For instance,
R B B R B R R B R
^ ^ ^
contains the length-3
same-color arithmetic progression shown.
Test cases
You'll probably only be able to test small cases.
+-----+-----+-----+-----+-----+-----+------+
| | k=1 | k=2 | k=3 | k=4 | k=5 | k=6 |
+-----+-----+-----+-----+-----+-----+------+
| r=1 | 1 | 2 | 3 | 4 | 5 | 6 |
| r=2 | 1 | 3 | 9 | 35 | 178 | 1132 |
| r=3 | 1 | 4 | 27 | 293 | | |
| r=4 | 1 | 5 | 76 | | | |
+-----+-----+-----+-----+-----+-----+------+
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