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 $k < \lfloor {n\over 4} \rfloor$ but does have a k ‐plex for every other $k \le {1\over 2} n$ . © 2008 Wiley Periodicals, Inc. J Combin Designs 16: 477–492, 2008