Dihedral and cyclic Ramsey numbers of the alternating 3-path

$R_{\mathrm{dih}}(P_3^{\mathrm{alt}}, K_b) = R_{\mathrm{cyc}}(P_3^{\mathrm{alt}}, K_b) = 2b - 1$ for all $b \in \mathbb{N}$ — the $a = 3$ slice of Conjecture 4.9 (Damnjanović–Đorđević, arXiv:2607.06817) and Conjecture 4.23 (Bašić–Damnjanović–Stevanović–Stošić, arXiv:2604.16188).

The a = 3 slice is settled outright. The parent conjecture's dihedral side has since been resolved for every a >= 4 as well (see the related entry), so Conjecture 4.9's claim 1 + (a-1)(b-1) now stands proved for all a >= 3; the trivial a = 1, 2 cases and the cyclic analogue for a >= 4 remain formally unaddressed.