Latin squares with no small odd plexes

A k-plex in a latin square of order n is a selection of kn entries in which each row, column and symbol is represented precisely k times. A transversal of a latin square corresponds to the case k =1. We show that for all even n > 2 there exists a latin square of order n which has no k-plex for any odd k < floor(n/4) but does have a k-plex for every other k up to n/2.

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