Same-color arithmetic progressions

Question

Van der Waerden's theorem says that

For any given positive integers r and k, there is some number N such that if the integers {1, 2, ..., N} are colored, each with one of r different colors, then there are at least k integers in arithmetic progression all of the same color. The least such N is the Van der Waerden number W(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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Answer 1

Python 3.5, 125 124 119 bytes

f=lambda r,k,*L:any(L[:x*k:x]==k*(*{*L[:x*k:x]},)for x in range(1,len(L)+1))*len(L)or max(f(r,k,i,*L)for i in range(r))

It's funny because, over the course of golfing this, the program actually got more efficient. Anything beyond f(2,4) or f(3,3) still takes forever, though.

Explanation

The initial version checked whether a sequence contained a progression of length k by trying all possible start indices and steps.

def f(r,k,L=[]):
 for i in range(len(L)):
  for j in range(len(L)):
   if len(set(L[i::j+1]))==1 and len(L[i::j+1])==k:
    return len(L)
 return max(f(r,k,L+[i])for i in range(r))

The golfed version only needs to try all possible steps since it prepends new sequence elements. The x*k cap is to take care of cases like [0, 0, 1], which contains a progression of length 2 but would not satisfy the uniqueness check uncapped.

As for the check

L[:x*k:x]==k*(*{*L[:x*k:x]},)

On the first pass of the golfed version, when L is empty, len(L) is 0. Thus the second half of the or will always be executed. After that L is non-empty, so {*L[:x*k:x]} (which is just Python 3.5 for set(L[:x*k:x])) will have at least one element.

Since L[:x*k:x] can have at most k elements and for L non-empty k*(*{*L[:x*k:x]},) has at least k elements, the two can only be equal when there are exactly k elements in both. For this to happen {*L[:x*k:x]} must have exactly one element, i.e. we only have one colour in our progression.