Condorcet domains are sets of linear orders with the property that, whenever voters' preferences are restricted to the domain, the pairwise majority relation (for an odd number of voters) is transitive and hence a linear order. Determining the maximum size of a Condorcet domain, sometimes under additional constraints, has been a longstanding problem in the mathematical theory of majority voting. The exact maximum is only known for $n\leq 8$ alternatives. In this paper we use a structural analysis of the largest domains for small $n$ to design a new inductive search method. Using an implementation of this method on a supercomputer, together with existing algorithms, we improve the size of the largest known domains for all $9 \leq n \leq 20$. These domains are then used in a separate construction to obtain the currently largest known domains for $21 \leq n \leq 25$, and to improve the best asymptotic lower bound for the maximum size of a Condorcet domain to $Ω(2.198139^n)$. Finally, we discuss properties of the domains found and state several open problems and conjectures.

翻译：孔多塞域是指满足以下性质的线性序集合：当投票者偏好被限制在该域内时，（对于奇数投票者）成对多数关系具有传递性，从而构成一个线性序。确定孔多塞域的最大规模（有时需附加约束条件）一直是多数投票数学理论中长期存在的问题。目前仅在备选方案数 $n\leq 8$ 的情况下已知确切最大值。本文通过对较小 $n$ 值最大域的结构分析，设计了一种新的归纳搜索方法。通过在超级计算机上实现该方法并结合现有算法，我们改进了 $9 \leq n \leq 20$ 范围内所有已知最大域的规模。随后利用这些域通过独立构造，获得了当前 $21 \leq n \leq 25$ 范围内的最大已知域，并将孔多塞域最大规模的最佳渐近下界提升至 $Ω(2.198139^n)$。最后，我们讨论了所发现域的性质，并提出了若干待解决问题与猜想。