AI 中文总结
研究偶数阶拉丁方横截线相交问题，构造特定拉丁方使每对横截线至少共享一个条目，推测不存在所有横截线共有的条目，通过算法证明该推测对\(n\leq10000\)成立，还研究了主导横截线的存在情况。
AI 中文摘要
阶为\(n\)的拉丁方是一个\(n\times n\)的阵列，其中\(n\)个符号中的每一个在每行和每列中恰好出现一次。拉丁方中的横截线是\(n\)个条目的选择，包括每行、每列和每个符号的一个代表。对于所有\(n\geq28\)的偶数阶（除\(n = 30\)外），我们构造了一个阶为\(n\)的拉丁方，其中每对横截线至少共享一个条目。我们推测在我们的拉丁方中不存在所有横截线都共有的单个条目。我们通过使用一种可能具有独立意义的算法找到横截线，证明了\(n\leq10000\)时的这个推测。我们称一条横截线是主导的，如果它与同一个拉丁方的所有其他横截线相交。我们表明，对于\(n\in\{5,7\}\)以及所有\(n\geq8\)且\(n\not\equiv3\bmod4\)的情况，存在具有主导横截线的阶为\(n\)的拉丁方。
英文摘要
A latin square of order $n$ is an $n\times n$ array in which each of $n$ symbols occurs exactly once in each row and column. A transversal in such a square is a selection of $n$ entries that includes one representative of each row and column, and one of each symbol. For all even orders $n\ge 28$ except $n=30$, we construct a latin square of order $n$ in which every pair of transversals share at least one entry. We conjecture that in our squares there is no single entry that is common to all transversals. We prove this conjecture for $n\le10\,000$ by finding transversals using an algorithm that is likely to be of independent interest. We say that a transversal is dominant if it intersects every other transversal of the same latin square. We show that there exist latin squares of order $n$ that have a dominant transversal for $n\in\{5,7\}$ and also for all $n\ge8$ such that $n\not\equiv3\bmod4$.
