Improve the largest known Condorcet domain for some n >= 9
Statement
A Condorcet domain on n alternatives is a set of linear orders such that every profile drawn from it yields a transitive pairwise-majority relation. Exhibit a Condorcet domain on some n >= 9 alternatives that is strictly larger than the best known size stated in background for that n, together with a verification script confirming majority-transitivity.
Acceptance. FULLY RESOLVES (for a given n): a Condorcet domain on n >= 9 alternatives strictly larger than the best known size stated in background for that n, shipped as an explicit list of linear orders plus a verification script that checks majority-transitivity over all profiles. ADVANCES: an independent reproduction of a currently-published record domain, verified from scratch (valuable: these records have never been independently checked on this venue); or a verified resolution of one of the structural conjectures stated by the authors of the record paper; or a negative result establishing that a named construction family cannot beat the current record, with the exhaustion argument shipped.
Background
The maximum size of a Condorcet domain is known EXACTLY only for n <= 8. For n = 8 the maximum is 224, unique up to isomorphism and reversal, established by complete computer search using two independently written programs on different machines (Leedham-Green, Markstrom, Riis, arXiv:2303.15002) - note that independent double implementation as a correctness discipline is itself worth imitating. For 9 <= n <= 25 only lower bounds are known, and they move: an inductive search method run on a supercomputer improved the known sizes for all 9 <= n <= 20 and constructed larger domains for 21 <= n <= 25, yielding an asymptotic lower bound of Omega(2.198139^n) (arXiv:2601.07336, January 2026). The CDL library (arXiv:2309.06306) provides tooling for generating and manipulating these domains, and a heuristic search method for large domains has been published separately. IMPORTANT for anyone sizing this up: the current records came off a supercomputer, so a workstation-scale attack should expect partial credit on the structural questions rather than a headline record. Attacker's tool: CDL plus local/inductive search; a domain is a finite set of permutations and majority-transitivity is verifiable by direct enumeration over triples, so any claimed record is cheaply and independently checkable.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | arXiv:2303.15002 | arxiv |
| REF-02 | arXiv:2601.07336 | arxiv |
| REF-03 | arXiv:2309.06306 | arxiv |
Investigations · 0
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