The van der Waerden theorem is a theorem in the branch of mathematics called Ramsey theory which states 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 whose elements are of the same color
This theorem applies to "classical" mathematics (which is basically mathematics based on classical logic). But are there any mathematical systems based on different types of logic (e.g non classical logics) where this theorem would not hold? Since there are logics where fundamental principles of classical logic do not hold (e.g the law of excluded middle in intuitionistic logic), wouldn't it be the same for this theorem?
Van der Waerden's theorem remains provable if you weaken classical logic to intuitionistic logic and if you weaken the usual axioms of arithmetic (or set theory) to just primitive recursive arithmetic. But the latter fact (about weakening arithmetic) comes from a nontrivial theorem of Shelah, which bounds the $N$ in the theorem by a primitive recursive function of $r$ and $k$.
If I remember correctly, Shelah's bound is at level $\mathcal E_5$ of the Grzegorczyk hierarchy, which essentially means iteration of iteration of exponentiation. So an ultrafinitist, who would be unwilling to accept that exponentiation is defined for all natural numbers (let alone its iteration and double-iteration) would not be convinced by the existing proofs of van der Waerden's theorem. I don't know anything about lower bounds for the $N$ in the theorem, but I'd be amazed if the $N$ could be bounded in a way that's ultrafinitistically acceptable; even an exponential bound would be big news.