Improved bounds on completion of partial Latin squares

A Latin square of order $n$ is an $n\times n$ array filled with $n$ symbols so that each symbol appears exactly once in every row and column. A partial Latin square of order $n$ is an $n\times n$ array whose cells are either empty or filled in such a way that each symbol appears at most once in every row and column, and at most $n$ distinct symbols are used. In 1983, Daykin and Häggkvist conjectured that every partial Latin square in which each row and column contains at most $n\mathrm{/}4$ symbols, and each symbol is used at most $n\mathrm{/}4$ times, can be completed to a Latin square. We prove that every partial Latin square in which each row and column contains at most $0.231n$ symbols, and each symbol is used at most $0.231n$ times, can be completed to a Latin square, significantly improving the previous best-known bound of $0.08n$, obtained by Fu and Weng. This problem can be seen as a partite analogue of the Nash-Williams conjecture concerning triangle decompositions of dense graphs, recently proved in the breakthrough work of Delcourt and Postle. Our proof uses a discharging' strategy, adapting the approach of Delcourt and Postle, combined with a novel method to achieve a balancedness’ property, required for the partite setting.