Skirting sets and the inverse football pool problem, with f(8,4) ≤ 36
Abstract
A tuple x ∈ ℤ_qⁿ skirts y ∈ ℤ_qⁿ if xᵢ ≠ yᵢ for every coordinate i, and S ⊆ ℤ_qⁿ is a skirting set if every tuple is skirted by some element of S; f(n,q) denotes the least size of a skirting set. This is the inverse football pool number T(n,q) of Östergård and Riihonen and of Brink, studied since the 1950s. It was reintroduced in 2026 by Adriaensen, Ihringer, Martin and Villagrán, who named skirting sets and identified f(n,q) with the total domination number of a direct power of a complete graph, and by Kuang and Wang as the number of binary subcubes needed to cover the ternary cube; neither 2026 paper cites the older work, and every interval in the tables they print is superseded by it. Already known are f(7,3)=29 and f(8,3)=44 (Brink), and f(6,4)=15 and f(7,4) ≤ 23 (Ashik Mathew and Östergård, under the name of blocking sets in a discrete torus). One cell is improved here: f(8,4) ≤ 36, against the published 40, so that the record for that cell becomes [27,36]. We also give explicit machine-checked witnesses at every cell of the 2026 tables — among them 44 tuples of ℤ₃⁸, 29 of ℤ₃⁷ and 15 of ℤ₄⁶ realising three of the known values above — each verified by complete evaluation from the definition, and three of them irredundant. We record besides that the growth constant C_q=inf_(n ≥ 1)f(n,q)^(1/n) equals q/(q-1) for every q ≥ 2, granted the Johnson–Lovász–Stein covering estimate as it has been printed for general q; the matching lower bound is a counting argument, and for q=3 no unproved estimate is needed. Every theorem below has been machine-checked in Lean 4.
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Archived files
- Version 1 · current (opens in a new tab)
Source snapshot 2026-08-30 15:34 UTC
File fingerprint
936495b32fd0d87715e6c13636fe258e8e9ae9df572bd028823e9438ddaa142e
Claim ledger
Stated results
K1routine2026-08-23
Skirting sets, the skirting number f(n,q), and the bridge from Finset (Fin n -> Fin q) to base-q codes
K2routine2026-08-23
Coordinatewise alphabet permutations, and the normalisation that puts both constant tuples in a smallest skirting set
K3routine2026-08-23
A set-cover search whose completeness is proved, and the lower-bound lemma it feeds
K4known2026-08-23
f(n,2) = 2ⁿ, and f(n,q) = n+1 for 0 < n < q
K5known data2026-08-23
The four table cells settled two-sided in the kernel: f(2,3)=3, f(3,3)=5, f(2,4)=3, f(3,4)=4
K6known data2026-08-23
f(4,3) = 8, the largest cell settled two-sided here
K7known2026-08-23
Kernel-checked witnesses for table cells the source states without a published witness (f(5,4) IS exhibited by its third displayed matrix – label corrected 2026-08-28; the unexhibited ones are f(5,3), f(6,3), f(4,4)): f(5,3) <= 12, f(6,3) <= 18, f(4,4) <= 7, f(5,4) <= 10, plus the source's own f(6,3) <= 21 construction
K8known2026-08-23
f(7,3) <= 29 – the source's bracket [28,30] narrowed to [28,29]
K9known2026-08-23
f(6,4) <= 15 – the source's bracket [13,16] narrowed to [13,15]
K11known2026-08-23
f(8,3) <= 45 – arXiv:2608.13252's bracket [41,50] narrowed to [41,45]
K10routine2026-08-23
Negative controls: neighbouring values, witness irredundancy, a successful search, no self-skirting, the normalisation is not vacuous, and q = 2
K12known2026-08-23
f(8,3) ≤ 44 — arXiv:2608.13252's bracket [41,50] narrowed to [41,44]; supersedes K11
K13routine2026-08-23
The 44-tuple witness is irredundant: all 44 single deletions fail
K14routine2026-08-23
The source's counting bound qⁿ ≤ f(n,q)(q−1)ⁿ (:163), proved, and C_q = inf_(n≥1) ⁿ√f(n,q) ≥ q/(q−1)
K15routine2026-08-23
C_q = q/(q−1) for every q ≥ 2 given the published Johnson–Lovász–Stein estimate — the source's closing Problem 1 (:417) answered, its Prop:Bounds bracket collapsed to its left endpoint
K16routine2026-08-23
The estimate beats ⁶√15 at a finite n: f(24,4) ≤ 48834 and 48834⁶ < 15²4
K17routine2026-08-23
Negative controls for the constant: the counting bound is strict at (3,3) and (3,4), the JLS hypothesis holds at f(4,3), C₄ > 1.33 unconditionally, C₄ < 1.5705 given JLS
K18known2026-08-28
f(7,4) ≤ 24 — the source's bracket [14,28] narrowed to [14,24], and its own ≤ 28 reproduced from its:133 product
K19candidate2026-08-28
f(8,4) ≤ 36 — the source's bracket [15,40] narrowed to [15,36]
K20routine2026-08-28
Controls for the two q = 4 cells: both witnesses break on deleting the first tuple, and both are irredundant
Provenance
- Generated by
- Machina Mathematica
- Released by
- Korea Superintelligence Labs
- Source context
- Source: Sam Adriaensen, Ferdinand Ihringer, William J. Martin, Ralihe R. Villagrán, *Skirting the n-tuples*, arXiv:2602.01080 (v1, 2026-02-01). Read from the paper's LaTeX source in the local arXiv corpus (/data/arxiv-text/mathₛ7ₚart₀024.txt, lines 1338558–1339001);:NNN below is a line number in that span re-based so:1 is the documentclass.
- Snapshot
- 2026-09-07 03:53 UTC
- Ledger commit
801848d7