Orthogonal latin rectangles
  We use a greedy probabilistic method to prove that for every $\epsilon > 0$, every $m\times n$ Latin rectangle on $n$ symbols has an orthogonal mate, where $m=(1-\epsilon)n$. 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. 
Roland H\"aggkvist and Anders Johansson