Digraph.main — ACₙACₙ is 5-ω̄-critical: Conjecture 5.10 at k = 5

Assembly of the k = 5 theorem (paper §3–§6). For every odd n = 2m+1 with m ≥ 3, the tournament T = ACₙACₙ:

• has ω̄(T) = 5 (omegabar_T5: k5_lower + k5_upper);
• has ω̄(T − v) = 4 for EVERY v (omegabar_T5_del: the bound at (0,0) spreads to all vertices by vertex-transitivity of the product);
• is hence 5-ω̄-critical (T5_kcritical5);
• has order n² (card_T5),

so distinct n give an infinite family of 5-ω̄-critical tournaments — Conjecture 5.10 of Aboulker–Aubian–Charbit–Lopes at k = 5 (conjecture_5_10_at_k5). Moreover every proper subtournament has ω̄ ≤ 4 (proper_sub_omegabar_le4), so the only subtournament with ω̄ ≥ 5 is T itself, of unbounded order: no bound ℓ(5) exists and Question 5.9 fails at k = 5 (question_5_9_fails_at_k5).

The product of the vertex-transitive ACₙ with itself is vertex-transitive.

ω̄(T) = 5

ω̄(T − v) = 4 for every vertex v

T is 5-ω̄-critical

T has order n²

Every proper subtournament has ω̄ ≤ 4

Conjecture 5.10 at k = 5: an infinite family

For every N there is a 5-ω̄-critical tournament on more than N vertices.

Question 5.9 fails at k = 5

No bound ℓ(5) exists: for every L there is a 5-ω̄-critical tournament on more than L vertices whose ONLY subtournament with ω̄ ≥ 5 is the whole tournament.