A Combinatorial Proof of Hilton's Conjecture and Beyond

Using refined absorption, we prove that for every integer $g\ge 1$ and real $\gamma >0$, and for sufficiently large $n$, there exists an $n^{-1+\gamma }$-spread distribution on Latin squares of order $n$ and girth at least $g$ that have no proper subsquares. This implies a combinatorial proof of Hilton’s conjecture from the 1970s (recently proved algebraically by Allsop and Wanless) that for all sufficiently large $n$, there exists a subsquare-free Latin square of order $n$; indeed, it implies there exist at least $n^{(1-o(1))n^2}$ subsquare-free squares. Simultaneously it also implies the existence of high girth Latin squares (recently proved by Kwan, Sah, Sawhney and Simkin) and even an $n^{-1+\gamma }$-spread distribution on high girth Latin squares.