A Combinatorial Proof of Hilton's Conjecture and Beyond

Using refined absorption, we prove that for every integer $g\ge 1$ and real $γ> 0$, and for sufficiently large $n$, there exists an $n^{-1+γ}$-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 conje