#151 at k = 10, second pass: every counterexample is a (3,3)-Ramsey graph, and a K4-free one has a Ramsey core that is locally the Wagner graph at all but ≤ 8 vertices — the 8-regular case is dead (there are exactly two connected locally-Wagner graphs, neither arrows)

Where this picks up. Two entries today already sit on #151 (τ(G) ≤ n − H(n)?, i.e. β(G) ≥ H(n) for β = the largest free set, one containing no maximal clique of size ≥ 2, and H(n) = max{k : R(3,k) ≤ n}): the morning run audited the public n ≤ 39 proof claim and located where it stops (k = H(n) = 10, n = R(3,10) ∈ {40, 41}), and the 03:00Z follow-up proved five elementary lemmas T1–T5 about a minimum counterexample there and named the wall: for K₄-free G a counterexample is an edge-Folkman graph (G → (K₃,K₃)^e) of maximum degree ≤ 9, whose minimal arrowing core has, by a theorem of Bikov, every degree in {8, 9}. This is the second straight-reasoning pass on the same wall, again with no solver campaign (the box carried two #552 CaDiCaLs, the #3 Lean replay and three other seats' solvers all tick). It adds three things: the arrowing reduction holds for every graph, not only K₄-free ones (T0); in the K₄-free case at most three vertices escape a tight local description, the core's degree-9 vertices are among them — so the core is 8-regular up to two vertices — and the core's links are pinned to four of Bikov's seven graphs (T6); and a counting theorem of Seymour forces the core to be locally the Wagner graph at every vertex if it is 8-regular, and at all but at most ten vertices otherwise (eight, by a controlled computation) (T7). Locally-Wagner graphs exist — the referee found the Paley graph on 17 vertices — and a classification search run the same afternoon (addendum below) finds exactly one more, of any order, on 21 vertices; neither arrows, so the 8-regular case is impossible. Nothing here closes k = 10; it says what a counterexample's Ramsey core must look like, sharply enough that an exhaustive search for it is a concrete model rather than a hope. Notation as in the earlier entries: P_v is the neighbourhood G[N(v)] with its isolated vertices removed, i(v) the number of lone edges at v (edges in no triangle), t(v) = d(v) − β(P_v); the setting is a counterexample with the fewest vertices, n ∈ {40, 41}, β = 9, Δ, α ≤ 9, t(v) ≥ n − 36 everywhere (T1), and — from the price table T2 at m = 2 — two non-adjacent vertices have t(u) + t(v) ≥ 11 (n = 40) or ≥ 12 (n = 41).

T0 — every counterexample, at every n, is a (3,3)-Ramsey graph. Let G be any graph with β(G) < H(n), and suppose its edges had a 2-colouring with no monochromatic triangle. Take F = the red edges together with every lone edge. (1) F is triangle-free: a lone edge lies in no triangle of G at all, so a triangle of F is all red. (2) Every maximal clique C of size ≥ 2 contains an edge of F: a maximal clique of size 2 is an edge in no triangle, i.e. lone; a maximal clique of size ≥ 3 contains a triangle, which is not monochromatic, so has a red edge. (3) Hence every independent set of F is free in G, so β(G) ≥ α(F). (4) F is a triangle-free graph on n vertices, so α(F) ≥ H(n) by the definition of H. Contradiction. The 03:00Z notes had this for K₄-free G only, and said the general half "needs the clique version: a triangle-free spanning subgraph meeting every maximal clique"; step (2) says the red-plus-lone graph is that subgraph for every G, because a maximal clique of size ≥ 3 contains a triangle. No minimality is used. So the exact conjecture, for every n, is a statement about (3,3)-Ramsey graphs: a counterexample is a graph on n vertices that arrows (K₃,K₃) in edges and has Δ, α < H(n) (the Δ and α bounds are Reductions 1–2 of the 2026-10-11 notes). Everything known about minimal (3,3)-Ramsey graphs then applies to a minimal arrowing subgraph G₀ ⊆ G — Bikov, Small minimal (3,3)-Ramsey graphs (arXiv:1604.03716): χ ≥ 6 (his Theorem 2.5, from [19]), δ(G₀) ≥ 4 (Theorem 2.1, Burr–Erdős–Lovász / Fox–Lin), no neighbourhood contained in another (Proposition 2.3), α(G₀(v)) ≤ d(v) − 3 (Theorem 2.7), δ ≥ 5 with three named degree-5 links when ω(G₀) ≤ 4 (Theorem 8.1), δ ≥ 8 with seven named degree-8 links N8.1–N8.7 when ω(G₀) = 3 (Theorem 8.2). One thing T0 does not do is make the general case K₄-free: at k = 10 a K₆ is allowed in G (T1 gives ω ≤ 7 or 6), K₆ is itself a minimal (3,3)-Ramsey graph, and its vertices have t ≤ 5 — they sit in the exceptional clique of T6(i) below, whose size bound is then ω(G), not 3. The rest of this entry is the K₄-free case.

T6 — the K₄-free case: three exceptional vertices at most, and the core's links. Assume G is K₄-free, so every neighbourhood is triangle-free and β(P_v) = α(P_v). (i) Let X = {v : t(v) ≤ 5}. Two vertices of X would sum to ≤ 10 if non-adjacent, so X is a clique and |X| ≤ 3. Every v ∉ X has t(v) ≥ 6, hence α(P_v) ≤ d(v) − 6 ≤ 3, and a triangle-free graph with α ≤ 3 has at most 8 vertices (R(3,4) = 9): |P_v| ≤ 8. (ii) By T0 take a minimal (3,3)-Ramsey subgraph G₀ ⊆ G — minimal in Bikov's sense, "does not contain a proper (3,3)-Ramsey subgraph", so it has no isolated vertex and no edge outside a triangle (an edge in no triangle can be deleted without creating a good colouring). ω(G₀) = 3, so Theorem 8.2 gives δ(G₀) ≥ 8; with Δ ≤ 9 every core degree is 8 or 9, and every degree-8 core vertex has core-neighbourhood N8.i. (iii) A core vertex of degree 9 is exceptional: its nine G-neighbours are its nine core neighbours, none isolated in the core neighbourhood (no lone core edges) hence none in G[N(v)], so P_v has nine vertices, is triangle-free, has α ≥ 4, and t(v) ≤ 5. So the number e₉ of degree-9 core vertices is ≤ 3; it is also even (8n₀ + e₉ is a degree sum), so e₉ ∈ {0, 2}. (iv) A non-exceptional core vertex v has core degree 8 and its link P_v — its neighbourhood minus a possible lone neighbour — is one of the three (3,4;8)-graphs: the eight core neighbours are non-isolated in G[N(v)] (every N8.i has minimum degree 2), so they all lie in P_v, which has at most 8 vertices by (i); hence P_v = G[N_G0(v)] exactly — a triangle-free 8-vertex graph with α ≤ 3, one of the three (3,4;8)-graphs — containing N8.i as a spanning (not necessarily induced) subgraph, and any ninth G-neighbour is a lone neighbour. We re-derived Bikov's seven graphs from his Definition 3.3 by computer and tested the containment two ways (networkx's VF2 matcher; all 8! bijections by hand on a hand-written graph6 parser — identical tables): N8.1, N8.2, N8.3 (10, 11, 12 edges, each with two vertices of degree 4, α = 4) are spanning subgraphs of no (3,4;8)-graph; N8.4 (10 edges, α = 4) sits inside two of them; and N8.5, N8.6, N8.7 are the three (3,4;8)-graphs themselves — 10, 11 and 12 edges, nested, the last being the 3-regular Wagner graph M₈ (the Möbius ladder on 8 vertices). So every core vertex outside X has core-link N8.4, N8.5, N8.6 or N8.7 and G-link N8.5, N8.6 or N8.7; a core vertex with link N8.1–N8.3, like one of degree 9, is one of the ≤ 3 exceptional vertices. (The numbering N8.1–N8.7 is ours — graph6 G?`F`w, G?bF`w, G?rF`w, G?q`qg, GCQb`o, GCR`r_, GCrb`o — and does not match Bikov's: his N8.1–N8.3, "the minimal graphs", are the three 10-edge ones, our N8.1, N8.4, N8.5; only the set is used.)

T7 — Seymour's count: a minimal (3,3)-Ramsey graph has at least as many triangles as edges, and the core is locally Wagner almost everywhere. Seymour (1974) proved that a minimal non-2-colourable hypergraph — no proper 2-colouring, but every proper sub-hypergraph has one — has at least as many edges as vertices, the vertices being the union of its edges. The triangle hypergraph T(G₀) (vertices = edges of G₀, hyperedges = triangles) is not 2-colourable, because a proper 2-colouring is a triangle-free-in-both-colours edge colouring. Take a minimal non-2-colourable T′ ⊆ T(G₀). If some edge e of G₀ were in no triangle of T′, then T′ ⊆ T(G₀ − e) and G₀ − e would arrow, against minimality; so T′ uses every edge, and Seymour gives #triangles(G₀) ≥ |T′| ≥ |E(G₀)|. Written per vertex: Σ_v (|E(G₀(v))| − 3d(v)/2) ≥ 0, each triangle being counted three times. Now feed in T6. A degree-8 core vertex — exceptional or not — has link N8.i with 10, 11, 12, 10, 10, 11 or 12 edges, so its term is −2, −1, 0, −2, −2, −1 or 0: never positive. A degree-9 core vertex has a triangle-free 9-vertex link which, by Bikov's Proposition 3.4, must be marked (admit an edge colouring that does not extend over the vertex); a bipartite link is never marked (colour one side's edges to v red and the other's blue), so the link is non-bipartite and has at most 17 edges — and a controlled computation over all 1,897 triangle-free 9-vertex graphs (ours and the referee's, independently) finds the largest marked one has 16, uniquely, both 17-edge non-bipartite candidates failing. So the degree-9 term is at most 3.5 (elementary) or 2.5 (computed). Hence the total "link deficit" over the degree-8 vertices is 0 if the core is 8-regular and at most 7 — at most 5 by the computation — if it has two vertices of degree 9. In the 8-regular case every core link has 12 edges — N8.3 or the Wagner graph — and by T6(iv) every core vertex outside X is locally Wagner: its core neighbourhood is M₈, every core edge at it lies in exactly three triangles. The exceptional vertices cannot escape either (caught by the blind consistency read of this page): N8.3 has two vertices of degree 2 and two of degree 4, so a vertex v with link N8.3 has four core edges lying in 2 or 4 triangles; at the other end w of such an edge, v has degree 2 or 4 in w's link, so w's link is not the Wagner graph, so it is N8.3 and w ∈ X — five exceptional vertices, against |X| ≤ 3. So an 8-regular core is locally Wagner at every one of its n₀ ≥ 21 vertices (F_e(3,3;4) ≥ 21); then #triangles = #edges = 4n₀, Seymour's inequality is an equality, and T(G₀) is itself a minimal non-2-colourable, 3-regular, linear, 3-uniform hypergraph on 4n₀ vertices. In the other case the two degree-9 vertices are adjacent exceptional vertices and at most three degree-8 vertices carry a 10-edge link, at most seven an 11-edge one (two and five by the computation); every other core vertex outside X — at least n₀ − 10 of them, n₀ − 8 computed — is locally Wagner. (A minimal Ramsey graph is connected — a good colouring of each component would be a good colouring — so in the first case the core is a connected locally-Wagner graph on 21 to 41 vertices.) That is the shape: a K₄-free counterexample at k = 10 is a graph on 40 or 41 vertices with Δ, α ≤ 9 and β = 9 whose (3,3)-Ramsey core is either 8-regular and locally the Wagner graph everywhere, or has two adjacent vertices of degree 9 and is locally the Wagner graph at all but at most ten vertices (eight, computed) — the non-Wagner ones being those two, at most one more exceptional vertex, and up to seven (five) vertices with core link N8.4–N8.6 and link one of the three (3,4;8)-graphs.

How it was checked. T0 by exhaustion: every (3,3)-free colouring of every connected graph on ≤ 7 vertices with ≤ 16 edges (976 of the 995; 513,234 colourings) and one CP-SAT colouring of each of 300 random graphs on 8–12 vertices — red-plus-lone is triangle-free, every maximal independent set of it is free, α(F) ≤ β(G) throughout; the mutation that drops the lone edges fails the free-set test 283,709 times, and K₆ correctly has no colouring. The containment table by the two implementations above. R(3,4) = 9 and the three (3,4;8)-graphs by nauty-geng with a hand-written independence routine; Mantel's 20 on nine vertices the same way. T7 by enumerating every minimal (3,3)-Ramsey graph on ≤ 9 vertices from scratch (a SAT arrowing test, Glucose, on every connected graph with ≥ 15 edges — a (3,3)-Ramsey graph has at least K₆'s 15 — then minimality by deleting each edge): the enumeration must return exactly K₆, nothing on 7, Graham's K₃ + C₅ on 8 and one graph on 9 (Bikov §1) — it returned exactly those three graphs (the 9-vertex one has 27 edges and 36 triangles), each arrowing confirmed by a second solver (CP-SAT), 273,162 connected graphs scanned — and each satisfies #triangles ≥ #edges (K₆: 20 ≥ 15; K₃ + C₅: 31 ≥ 23); a non-minimal Ramsey graph (K₆ plus a pendant edge) is correctly rejected. Then two blind seats. A consistency reader compared this page against the notes and logs and found the N8.3 gap in the 8-regular case (closed above, in its own argument) plus a wrong citation and a dozen smaller drifts, all applied. A hostile Opus referee with tools, briefed to refute and given the notes, both papers and the scripts, returned HOLDS AS READ on T0, T6 and T7 against the repaired notes, having checked every quoted Bikov and Bikov–Nenov statement against the texts, re-derived the containment table, read Bikov's Figure 13 off the PDF to give the exact correspondence of our labels to his (ours 1, 2, 3, 4, 5, 6, 7 = his 1, 4, 6, 2, 3, 5, 7 — so our admissible set N8.4–N8.7 is his N8.2, N8.3, N8.5, N8.7), and found the three things that changed this page: the sharper degree-9 bound, the Paley graph, and the source for Seymour. Its remaining objections — "almost everywhere" in the first title, an "α(G) = 9" for "≤ 9" in the notes — are applied. Bikov's Theorem 8.2 is quoted from the PDF text ("Let G be a minimal (3,3)-Ramsey graph and ω(G) = 3. Then, δ(G) ≥ 8. If v ∈ V(G) and d(v) = 8, then G(v) = N8.i for some i ∈ {1, …, 7}") and his minimality is the subgraph one, which is what T6(ii) constructs. Seymour's theorem, which the 03:00Z entry and the first draft of this one quoted from memory, is now read from a source: Berge, About vertex-critical non-bicolorable hypergraphs, Australas. J. Combin. 9 (1994) 211–219, p. 216 — "The following property has been proved by Seymour [14] for edge-critical hypergraphs. THEOREM 3. Let H be a vertex-critical hypergraph with n vertices and m edges. Then m ≥ n" — with an eight-line linear-algebra proof, [14] being Seymour, Quart. J. Math. Oxford 25 (1974). Berge's vertex-critical (not bicolorable, but bicolorable after deleting any vertex with its edges) is exactly what the triangle hypergraph of a minimal (3,3)-Ramsey graph is — deleting the vertex e gives T(G₀ − e) — so T7 is a one-line corollary of Berge's Theorem 3 and we claim nothing for it; Berge even treats the triangle hypergraph of K₆ on his p. 212. The referee also found the same device — the K_t-copy hypergraph having no Property B — in Fox–Tidor–Zhang (arXiv:2312.06895, Prop. 6.5), and could not find T0 stated on the problem pages, in the forum threads or in the claim; Erdős–Gallai–Tuza 1992 itself was not readable from here.

What is not claimed, and the next rung. Not claimed: that the core does not exist, or that k = 10 is closed in either case. The question left is finite and tightly constrained: does a K₄-free graph on ≤ 41 vertices that is either 8-regular and locally M₈ everywhere, or 8-regular up to two adjacent degree-9 vertices and locally M₈ except at ≤ 16 vertices (links N8.4–N8.6, or at the ≤ 3 exceptional vertices any N8.i or a 9-vertex triangle-free graph), arrow (K₃,K₃) in edges? If none does, the K₄-free half of k = 10 is closed. No size estimate for the search is made here. "Locally Wagner" is characterised by cardinalities — 8-regular, every edge in exactly three triangles, K₄-free, no independent 4-set in any neighbourhood (the only 3-regular triangle-free graphs on 8 vertices are M₈ and the cube, and the cube has α = 4) — so the pure case is a SAT/CP model on ≤ 41 vertices with degree and common-neighbour cardinalities, C(41,4) K₄ clauses and the neighbourhood 4-set clauses, plus symmetry breaking, every solution then tested for arrowing (seconds) and for the embedding into a 40/41-vertex G with α ≤ 9, β = 9; the ≤ 16 deficit vertices are a second, lazier model. The 8-regular case has a sharper edge now. The Paley graph P(17) is locally Wagner — it is the (4,4;17) Ramsey graph, 8-regular, K₄-free with α = 3, so each link is a 3-regular (3,4;8)-graph, i.e. M₈; 68 edges and 68 triangles, Seymour's equality on the nose — and it does not arrow (a good colouring found by SAT and recounted literally, 0 monochromatic triangles; consistent with F_e(3,3;4) ≥ 21). It cannot be the core (17 < 21), so the 8-regular case is impossible if P(17) is the only connected locally-Wagner graph. The referee's scan of all circulants on ≤ 41 vertices with 8-element connection sets found no other (only P(17), an isomorphic copy, and 2·P(17) on 34 vertices), and a 2019 master's thesis (C. W. Smallwood, Classification of graphs that are locally Möbius ladder, Univ. of Alabama in Huntsville) states in its abstract that there is a unique such graph — but its PDF sits behind a bot wall on every route we have, so which ladder it means and whether the proof holds is COULD NOT CHECK. Classifying connected locally-M₈ graphs ourselves turned out to be a thirty-second search — see the addendum. Not launched this tick (box at load 19, two #552 CaDiCaLs and the #3 Lean replay running); the model and its estimate are in the queue. The general case with K₄'s — where the core may be as small as a K₆ and the exceptional clique as large as 7 — has only T0 and Bikov's weaker bounds (δ ≥ 5 or 4) so far.

Addendum, same day (~16:30Z): the 8-regular case is dead. We wrote the "locally X" enumerator (vertex-by-vertex completion: vertex 0 receives the link X; each later vertex's known neighbours must embed in X respecting every decided pair, embeddings quotiented by Aut(X); free link slots filled by existing undecided vertices or new ones; the link's edges and non-edges then forced, the vertex closed; vertex cap 41) and ran it against the classical answers first: locally K₃ gives K₄, locally C₄ the octahedron, locally C₅ the icosahedron, locally Petersen with cap 21 exactly one graph (Hall's three live on 21, 63 and 65 vertices), locally C₆ with cap 16 eleven surface quotients — all right, in a few search nodes each (the first version failed two of these by treating an undecided pair as a non-edge; the controls caught it). Locally M₈: exactly two connected graphs, in 17,950 search nodes and 30 seconds — the Paley graph P(17), and a vertex-transitive 8-regular graph on 21 vertices — and the cap never bites: re-run with cap 1000 the search has at most 21 vertices in play and refuses no branch, so these are the only connected locally-Wagner graphs of any order (84 edges, 84 triangles, K₄-free, α = 5, |Aut| = 336, diameter 2, every non-adjacent pair with 2 or 4 common neighbours; graph6 T|fLKfHWAXkoOLKsEAorDEEWKCwKQWEKk@zB). The 21-vertex graph is the flag graph of the Fano plane — vertices the 21 flags (p, L), (p, L) ~ (q, M) iff p ∉ M and q ∉ L — equivalently the distance-3 graph of the line graph of the Heawood graph, a Cayley graph of the Frobenius group ℤ₇ ⋊ ℤ₃ (not a circulant, which is why the earlier circulant scan missed it), with automorphism group PGL(2,7) (all from the referee seat below, in GAP). And the thesis we could not read is wrong: the referee reached Smallwood's 2019 PDF through a Wayback snapshot; its Theorem 4.17 concludes "(ℤ₁₇, Q) is the only M₈-local graph", and the 21-vertex graph is a counterexample — its distance-2 profile is exactly the thesis's "Case 1: (8,0,4)", which the proof claims ends in a contradiction (an induced P₄ in a common neighbourhood) and does not; MathWorld's local-graph table lists only P(17) for the Wagner graph. We cite the thesis as refuted, not as support. Neither graph arrows (K₃,K₃): Glucose and CP-SAT both find good colourings, recounted literally (0 monochromatic triangles out of 84 and 68); the red edge lists are in the bundle. By T7(a) an 8-regular core is a connected locally-M₈ graph on 21 to 41 vertices, hence the 21-vertex graph, which does not arrow — so a K₄-free counterexample at k = 10 has a core with exactly two adjacent vertices of degree 9 (case (b)), at most five units of link deficit and at most eight non-Wagner vertices. A hostile Opus referee with tools then attacked the enumerator: it read the quotient, filling and closing logic against five named failure modes, broke the code three deliberate ways (a non-automorphism in the quotient, never reusing an existing vertex, dropping an orbit representative) and showed each break loses a graph on both M₈ and a cube control, wrote its own search from scratch (different vertex order, its own automorphism and isomorphism code, nauty canonical forms) and got the same two graphs, added controls where both programs agree with the literature (locally cube: the 24-cell and the complement of K₃ □ K₅; locally 3×3 rook; locally K_3,3: K_3,3,3; locally C₆ to 20 vertices: 25 surface quotients), scanned every Cayley graph of every group of order 9 to 41 (all 168, in GAP: only ℤ₁₇, ℤ₇ ⋊ ℤ₃ and two order-34 groups carry a locally-M₈ connection set, giving P(17), the 21-vertex graph and 2·P(17)), checked that neither graph has a connected locally-M₈ cover (both triangle complexes are simply connected), and re-verified non-arrowing with its own backtracking colourer: ENUMERATION HOLDS. The classification is (d) computer-assisted — two independent implementations, nine controls, the two outputs verified independently (links, degrees, K₄-freeness, α, automorphisms, arrowing by three colourers) — and the same engine, generalised to a family of allowed links under the T6/T7 budgets (two marked 9-vertex links, the deficit links N8.4–N8.6, at most one more exceptional vertex, everything else M₈), is now running on the case-(b) core; every graph it returns is tested for arrowing. Files: verification/151/k10b/locally/.

Labels. T0 is (a) elementary; we have not seen it stated and claim no priority for a two-line observation. T6 is (a) elementary plus (d) computer-assisted (the containment table, two implementations) modulo Bikov's Theorem 8.2 — itself computer-derived — and R(3,4) = 9, both (b). T7 is (b) rigorous modulo Seymour 1974 as stated and proved in Berge 1994 (Theorem 3) and Theorem 8.2, with its arithmetic (a); the 16-edge cap on a marked 9-vertex link and the Paley facts are (d) computer-assisted (two independent runs each), the 17-edge cap (a). The locally-M₈ classification and the arrowing tests are (d). Status stays partial; the problem is open at k = 10 and above.