# Monogamous Latin squares

We show for all n not in {1,2,4} that there exists a latin square of
order n that contains two entries γ_{1} and
γ_{2} such that there are some transversals through
γ_{1} but they all include γ_{2} as
well. We use this result to show that if n>6 and n is not of the form
2p for a prime p≥11 then there exists a latin square of order n that
possesses an orthogonal mate but is not in any triple of MOLS. Such
squares provide examples of 2-maxMOLS.
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