Dihedral Ramsey numbers of the alternating a-path versus K_b, for every a >= 4: 1 + (a-1)(b-1)

$R_{\mathrm{dih}}(P_a^{\mathrm{alt}}, K_b) = 1 + (a-1)(b-1)$ for all $a \geq 4$, $b \geq 1$ — the $a \geq 4$ slice of Conjecture 4.9 (Damnjanović–Đorđević, arXiv:2607.06817). Combined with the $a = 3$ case (see sibling entry), this resolves Conjecture 4.9 in full for $a \geq 3$.

The dihedral case only, for every $a \ge 4$ and $b \ge 1$; the substance is the upper bound, which the source paper's own computations could not reach. Together with the sibling a = 3 entry this proves Conjecture 4.9's claim $1+(a-1)(b-1)$ for all $a \ge 3$; the conjecture's trivial a = 1, 2 cases are unaddressed by either entry, and the cyclic analogue $R_{cyc}(P_a^{alt}, K_b)$ for $a \ge 4$ remains open. The engine is a self-contained inequality of independent interest: for any graph on a linearly ordered vertex set, the alternating-path reach statistics satisfy $\sum_m [P(m)+Q(m)] \ge 2|E(G)|$, from which the theorem falls out by averaging and a pivot decomposition.