  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 will
investigate the class of \leq\epsilon-dense partial latin squares: partial
latin squares in which each symbol, row, and column contains \leq\epsilon
n-many nonblank cells. A conjecture of Nash-Williams on triangulations of
graphs led Daykin and H\"aggkvist to conjecture that all \leq(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
\leq\epsilon-dense partial latin squares that contain \leqdn^2 cells.
  In particular, we establish that all \leq(1/5300)-dense n by n partial latin
squares are completable, as well as all \leq(1/13)-dense n by n partial latin
squares that contain \leq(8.8*10^(-5)*n^2)-many filled cells. This improves
prior results of Gustavsson, which required \epsilon = d \leq 10^(-7), as well
as Chetwynd and Haggkvist, which required \epsilon = d \leq 10^(-5) and n even,
\leq 10^7.
