[{"data":1,"prerenderedAt":4},["ShallowReactive",2],{"fEnVVbSqOO":3},"# A 7-vertex tournament whose Sperner capacity exceeds the order of its largest transitive subtournament\n\nFor a digraph `R` on a finite vertex set, `w(Rⁿ)` is the largest number of words of length `n` over\nthe vertices such that every ordered pair of distinct words has a coordinate carrying an arc from\nthe first to the second, and the capacity is `C(R) = lim w(Rⁿ)^{1/n} = sup_n w(Rⁿ)^{1/n}`\n(Alon, *On the capacity of digraphs*, European Journal of Combinatorics 19 (1998) 1–5; the Sperner\ncapacity of Gargano, Körner and Vaccaro is its logarithm). For a tournament `T`, `t(T)` is the number\nof vertices of its largest transitive subtournament, and `t(T) ≤ C(T)` always. Conjecture 1.1 of\nAlon's paper, attributed there to Körner and Simonyi, reads `C(T) = t(T)` for every tournament; the\nsame paper disproves it, by a random construction and by the Paley tournament on 67 vertices, and\nasks for the smallest tournament for which the equality fails. The answer is seven.\n\nLet `T₇` be the subtournament of the Paley tournament on 23 vertices (`x → y` iff `y − x` is a\nnonzero square modulo 23) induced on the residues 0, 1, 2, 3, 9, 14, 18. [Challenge.lean](Challenge.lean)\nstates nineteen theorems over nine definitions and [Solution.lean](Solution.lean) proves them,\nkernel-only; the table in [VERIFICATION.md](VERIFICATION.md) names each one:\n\n- `T₇` is a tournament with `t(T₇) = 4`; eighteen listed points of `T₇ × T₇` form a transitive\n  clique of its second Sperner power (every earlier point sends an arc to every later one in some\n  coordinate), so by the lift lemma some Sperner power of `T₇` has a clique larger than `4ⁿ`,\n  `√18 ≤ C(T₇)`, and `t(T₇) \u003C C(T₇)`: the equality fails on seven vertices;\n- `C(T₇) ≤ 5`, by an explicit rank-5 factorization over `GF(2)` of a matrix with nonzero diagonal\n  and zeros on the arcs (the rank bound);\n- every tournament on at most six vertices satisfies `C(T) = t(T)`: for each of the 1, 1, 1, 2, 4,\n  12, 56 isomorphism classes on 0 to 6 vertices a representative with a rank-`t` factorization over\n  `GF(2)` and a transitive `t`-chain, and a kernel check that every labelled tournament on at most\n  six vertices is a relabelling of a representative; hence a tournament with `t(T) \u003C C(T)` has at\n  least seven vertices, and seven is exactly the least order of a counterexample;\n- the general tools: the lift lemma, `N^{1/k} ≤ C(R)` for `N` such points of the `k`-th power,\n  `t(R) ≤ C(R)`, the rank bound `C(R) ≤ r` for any factorization through `Fʳ`, and `C(R) ≤ |V|`;\n- the Paley tournament on 23 vertices has `t = 5` and `√29 ≤ C`, from a 29-point transitive clique\n  of its second Sperner power.\n\nOn paper, in [note/](note/README.md), with standard-library certificates: the square value 18 is\nexact; on seven vertices, 455 of the 456 isomorphism classes have a rank-`t` matrix over `GF(2)` and\nthe exception is `T₇`, which is therefore the unique smallest counterexample up to isomorphism; every\nproper subtournament of `T₇` satisfies the equality; `√18 ≤ C(T₇) ≤ 5`; the Paley tournament on 7\nvertices has `C = 3`. The smallest example previously exhibited has 67 vertices; examples on 27 and\n26 vertices follow from Alon's argument over `GF(27)` with a published value of `t`, and are\nre-verified in the note.\n\nNot claimed: the exact value of `C(T₇)`; in Lean, anything about tournaments on seven or more\nvertices other than `T₇` and the Paley tournament on 23 vertices (the uniqueness of `T₇` and the\nexact square value are certificates, not Lean theorems).\n\n> As of 2026-10-08 (UTC), no tournament on fewer than 67 vertices whose capacity exceeds its\n> transitive number, no determination of the capacity of every tournament on six vertices, and no\n> statement that the smallest counterexample has seven vertices or is unique, was located in Alon's\n> paper and his survey *Graph powers*, Körner's 1998 paper, Kiviluoto–Östergård–Vaskelainen's\n> *Sperner capacity of small digraphs* (which settles every digraph on at most five vertices except\n> eight that are not tournaments), Vaskelainen's thesis, Simonyi's 2006 dissertation, the arXiv\n> literature on Sperner capacity, zbMATH, Google Scholar, the Palomar registry, Hexagon, openai/math\n> or the formal-conjectures repository. The lift lemma and the rank bound are standard in substance\n> (Sali–Simonyi 1999; Kiviluoto–Östergård–Vaskelainen 2009, Theorem 4); what is new is their values\n> on `T₇` and the classification of the small tournaments.\n\nLean `v4.35.0-rc2` and Mathlib `v4.35.0-rc2` (commit `065356127b1dc0016f66b7283ce0ce2c4055aa55`) are\npinned by the committed manifest; there are no GitHub Actions workflows.\n\n```sh\npython scripts/verify.py --fetch-cache\n```\n\nruns every check (the pins, the source guard, the definition and statement comparisons, the\ncertificates, the build with the axiom audit, the module-resolution check, the elaboration check of\nthe nine definitions against the Challenge, and Palomar's core-notation audit);\n[VERIFICATION.md](VERIFICATION.md) lists them and their limits. [PROOF.md](PROOF.md) gives the\nmathematics with the Lean name of every step, [DISCLOSURE.md](DISCLOSURE.md) the assistance\nstatement, and [note/](note/README.md) the research note (CC BY-SA 4.0) with its certificates.\n\nLicense: [MIT](LICENSE).\n",1791742520816]