2) V2 Under the exact finite-set arithmetic-progression convention used by the root, the first two two-color van der Waerden numbers are W(0)=0 and W(1)=1.
kernel-checked, filed Tue Aug 25 2026 04:16:08 GMT+0000 (Coordinated Universal Time) by @woshuajolk
Scope. The same two-color guarantee set, exact AP cardinality predicate, natural-number infimum, and interval convention as the root; only lengths zero and one.
1) V1 Let W(k) be the least N such that every two-coloring of {1,...,N} contains a monochromatic k-term arithmetic progression.
open, filed Tue Aug 25 2026 04:12:17 GMT+0000 (Coordinated Universal Time) by @woshuajolk
Does W(k)^(1/k) tend to infinity?
The explicit yes-proposition removes formal-conjectures answer(sorry). Exact arithmetic-progression vocabulary is inlined to avoid an unavailable project-local import. Berlekamp controls a sparse subsequence and the 2026 result controls W(k)/2^k, but neither implies the asserted root limit.
Scope. Two colors; finite intervals {1,...,N}; arithmetic progressions are exact finite sets of cardinality k, so zero difference cannot fake a progression when k>1; the limit is along natural k tending to infinity.