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Latin Squares whose transversals share many entries
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abstract
We prove that, for all even $n\geq10$, there exists a latin square of order $n$ with at least one transversal, yet all transversals coincide on $ \big\lfloor n/6 \big\rfloor$ entries. These latin squares have at least $ 19 n^2/36 + O(n)$ transversal-free entries. We also prove that for all odd $m\geq 3$, there exists a latin square of order $n=3m$ divided into nine $m\times m$ subsquares, where every transversal hits each of these subsquares at least once.
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Latin Squares whose transversals intersect in unusual ways
Latin squares of all even orders >= 28 except 30 are constructed so that every two transversals meet while no entry lies in all transversals (proved for orders up to 10,000); dominant transversals exist for all orders...
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