Tournaments not inducible by five voters

Abstract

A tournament T is k-inducible if there are k linear orders on its vertex set such that, for every arc $i \to j$ of T, a majority of the orders rank i above j. For odd k, let N(k) be the least order at which some tournament is not k-inducible. Only N(3) = 8 is known exactly; for N(5) the best bounds were $12 \le N(5) \le 38$, from our previous paper [2], which also gave the first explicit example of moderate order, the Paley tournament $P_{43}$. Results. A bespoke search algorithm improves both ends: $13 \le N(5) \le 23$. The upper bound comes from proving that $P_{23}$ is not 5-inducible, the case Bachmeier et al. [1] reported they could not decide, their SAT solver not having terminated within a cumulative six weeks; ours takes 22 hours on one laptop. The lower bound comes from an analysis at order 12. We also show that $P_{31}$ is not 5-inducible, while $P_{19}$ is 5-inducible but not with unit margin, that is, not by a profile in which every arc is carried by exactly three voters against two. Both $P_{19}$ and $P_{23}$ are arc-critical for their respective properties, whereas $P_{31}$ and $P_{43}$ are not vertex-critical: deleting a vertex leaves a tournament that is still not 5-inducible. Method. The search places one vertex at a time, always choosing the vertex with the fewest options left, and propagates the consequences. Together with the automorphisms of the tournament, this decides on a single laptop instances that neither integer programming nor a general-purpose SAT solver can settle. The refutations for $P_{19}$ and $P_{23}$ are certified as well: the search is split into independent subproblems, a SAT solver emits a machine-checkable proof for each, and a separate program rechecks every proof. All results, subject to two human-checked lemmas, are reproducible from https://github.com/Leonardini/TournamentsBeyond5Voters.