Abstract. A classical question in combinatorics is the following: given a partial latin square P , when can we complete P to a latin square L? In this paper, we investigate the class of -dense partial latin squares: partial latin squares in which each symbol, row, and column contains no more than n-many nonblank cells. Based on a conjecture of Nash-Williams, Daykin and Häggkvist conjectured that all 1 4 -dense partial latin squares are completable. In this paper, we will discuss the proof methods and results used in previous attempts to resolve this conjecture, introduce a novel technique derived from a paper by Jacobson and Matthews on generating random latin squares, and use this technique to study -dense partial latin squares that contain no more than δn 2 filled cells in total.In this paper, we construct completions for all -dense partial latin squares containing no more than δn 2 filled cells in total, given that < 1 12 10409. In particular, we show that all 9.8 · 10 −5 -dense partial latin squares are completable.These results improve prior work by Gustavsson, which required = δ ≤ 10 −7 , as well as Chetwynd and Häggkvist, which required = δ = 10 −5 , n even and greater than 10 7 .

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