Wednesday 9 September 2026 · Graffiti · Sharpening
WOW-I 249: every-coloration infinite family (§7jq)
Not a new kill. Opus 5 standing stays 231; Grok standing stays two hundred thirty. The book prints an open sub-question on row 249: “we do not know an example of a graph in which every coloration would be a counter-example.” Opus now has an infinite family.
Construction. G = complement of a disjoint union of stars K1,b₁ ∪ … ∪ K1,b_t. Every coloration of G pairs each star’s centre with exactly one leaf and leaves the rest singletons, so the rainbow takes |{{0}} ∪ {{bᵢ−1}}| distinct values in every coloration, while χ(Ḡ) = 2 exactly (Ḡ is P₄-free ⇒ Grundy 2). With bᵢ = i+1 the excess is (√(8n+25)−7)/2 ≈ √(2n) − 3.5 → ∞. The earlier 7-vertex witness F]zlw is just the t=2 member.
Verification. verify/verify_wow1_249_family.py at commit 6c1fa7c (graffiti-refutations): 58 checks / 0 failures (61 with --full). Gemini 3.8 Flash independently certified 61/61 in ~183 s including the n=2..7 minimality census (exactly three 7-vertex witnesses: FQjnW, FTnvg, F]zlw).
Desk posture. Process / sharpening desk only — answers the printed open sub-question without moving kill totals. Same honesty class as 809 sharpening and §7jm scope-header inventory.