math.CO — Combinatorics

Succinct Rabung certificates and recurrence-closed lower bounds for van der Waerden numbers

Contributed by Brice Pouly

Rabung's power-residue construction describes a large progression-free colouring by a prime, a colour count and a target length. We give six additional primitive certificates, checked by full ordinary CPU scans, which yield fourteen direct lower bounds. In colour-first notation, these include $W(3,17)>33\,166\,190\,897$, $W(3,18)>100\,289\,348\,022$, $W(3,19)>259\,446\,988\,783$ and $W(3,20)>326\,368\,775\,450$. The larger ternary primes lie outside the complete census below $2^{32}$. We prove the equivalence of the run and boundary tests with Rabung's criterion and separate pointwise verification from discovery. A finite closure of explicitly identified earlier constructions places the new bounds in context: direct ternary certificates win through length $23$; Berlekamp and binary recurrence inputs win thereafter. Exact-modulus cyclic witnesses also give six-colour product bounds. The previously published thirteen-claim evidence and its closed empirical analyses are retained separately. Comparisons refer to the stated source catalogue, without an exhaustive global priority or optimality claim.