Completions of epsilon-dense partial Latin squares
  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. 
Padraic Bartlett
math.CO