2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/124724We use a greedy probabilistic method to prove that for every $ε> 0$, every $m\times n$ Latin rectangle on $n$ symbols has an orthogonal mate, where $m=(1-ε)n$. That is, we show the existence of a second Latin rectangle such that no pair of the $mn$ cells receives the same pair of symbols in the two rectangles.Accepted for publication in CPCCombinatorics05B15text