A partial transversal T of a Latin square L is a set of entries of L in which each row, column and symbol is represented at most once. A partial transversal is maximal if it is not contained in a larger partial transversal. Any maximal partial transversal of a Latin square of order n has size at least $\lceil \frac{n}{2}\rceil$ and at most n. We say that a Latin square is omniversal if it possesses a maximal partial transversal of all plausible sizes and is near-omniversal if it possesses a maximal partial transversal of all plausible sizes except one.

Evans (2019) showed that omniversal Latin squares of order n exist for any odd $n\ne 3$. By extending this result, we show that an omniversal Latin square of order n exists if and only if $n\notin \{3,4\}$ and $n\not\equiv 2(\mathrm{mod}4)$. Furthermore, we show that near-omniversal Latin squares exist for all orders $n\equiv 2(\mathrm{mod}4)$.

Finally, we show that no non-trivial finite group has an omniversal Cayley table, and only 15 finite groups have a near-omniversal Cayley table. In fact, as n grows, Cayley tables of groups of order n miss a constant fraction of the plausible sizes of maximal partial transversals. In the course of proving this, we partially solve the following interesting problem in combinatorial group theory. Suppose that we have two finite subsets $R,C\subseteq G$ of a group G such that $|\{rc:r\in R,c\in C\}|=m$. How large do $|R|$ and $|C|$ need to be (in terms of m) to be certain that $R\subseteq xH$ and $C\subseteq Hy$ for some subgroup H of order m in G, and $x,y\in G$?

甲局部横向Ť拉丁方的大号是一组条目的大号，其中每一行，列和符号被至多一次表示。如果部分横向不包含在较大的局部横向中，则该局部横向最大。n阶拉丁方的任何最大局部横截面的大小至少为$\lceil \frac{ñ}{2}\rceil$最多n个。我们说一个拉丁方是omn​​iversal，如果它拥有所有合理尺寸的最大部分横向和是近omn​​iversal如果它拥有所有合理尺寸的最大部分横向只有一个除外。

埃文斯（Evans，2019）表明，存在所有n阶的全拉丁方格$ñ\ne 3$。通过扩展此结果，我们表明存在且仅当存在以下情况时，阶n的全拉丁方格存在$ñ\notin \{3，4\}$ 和 $ñ\not\equiv 2（\mathrm{模}4）$。此外，我们表明所有订单都存在近乎全能的拉丁方格$ñ\equiv 2（\mathrm{模}4）$。

最后，我们证明没有非平凡的有限群没有全能的Cayley表，只有15个有限群具有近乎全能的Cayley表。事实上，作为Ñ增长，顺序组的凯莱表Ñ错过最大局部断面的合理大小的恒定部分。在证明这一点的过程中，我们部分地解决了组合群论中的以下有趣问题。假设我们有两个有限子集$\mathrm{[R}，C\subseteq G$一组ģ使得$|\{\mathrm{[R}C：\mathrm{[R}\in \mathrm{[R}，C\in C\}|=米$。多大$|\mathrm{[R}|$ 和 $|C|$需要（根据m）确定$\mathrm{[R}\subseteq XH$ 和 $C\subseteq Hÿ$对于一些亚组ħ的顺序米在ģ，和$X，ÿ\in G$？