Monochromatic 4-AP avoidance in 2-colorings of Z/pZ for primes p >= 5 and a computation of W_c(4,2)

Abstract

We study 2-colorings of Z/pZ that avoid monochromatic 4-term arithmetic progressions for every step d with p not dividing d. We prove a complete classification for primes: such a coloring exists if and only if p is in {5, 7, 11}. When solutions exist, the minimal period equals p, and we enumerate them up to dihedral symmetries and a global color swap. Nonexistence for all other primes combines DRAT-verified UNSAT certificates for 13 ≤ p ≤ 997 with a cyclic van der Waerden corollary that forces nonexistence for every prime p ≥ 34. Using the same SAT/DRAT pipeline on composite moduli (restricted to non-degenerate windows), we certify the exact cyclic van der Waerden value W_c(4,2) = 34: we find a witness at M = 33 and produce a DRAT-verified UNSAT certificate at M = 34. For all M ≥ 35 the bound W_c(4,2) ≤ W(4,2) = 35 implies unavoidability. All scripts and proof logs are provided for exact reproduction.