A partial Latin squares problen1 posed by Blackburn

Abstract

Blackburn asked for the largest possible deusity of fillf:d cells in a pa rtial latiu square with Llw property tha t, whenever t,w > distinct cells Pn.1, and Pci1 are occupied by t.he same symbol the 'opposite corners' P,,,, and Pbc are blank. \Ve show that, as the order n of the partial lat,iu square increases, a density of at least exp (-c(logn) L/'.! ) is possible using a diagonally cyr.lic constmction, where c is a. positive constant. The q uestion of whether a constant density is achievable remains, b ut we show that a density exceeding¼ ( v'fI -1)(1 + 4/ n) is not possible.