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Claim. Theorem 2 of Erdős and Kaplansky's paper states that the number $L_{k,n}$ of $k\times n$ Latin rectangles with labeled rows, columns and symbols satisfies

$$L_{k,n}\sim e^{-\binom k2}(n!)^k$$

as $n\to\infty$ whenever $k<(\log n)^{3/2-\epsilon}$ for a fixed $\epsilon>0$. Theorem 1 is the one-row step: in that range the number of rows that extend a given $k\times n$ rectangle is $n!\,e^{-k}$ up to a relative error $O(n^{-\delta})$, and the asymptotic follows by multiplying the steps. The method is a double inclusion-exclusion, over the columns in which a candidate row clashes with the rectangle and over repeated pairs of equal symbols. The authors note that $(\log n)^{3/2}$ appears to be a natural boundary of the method and say they believe the actual break occurs at $k=n^{1/3}$; the sketched expansion of Section 4 suggests, without proof, that the formula ceases to be valid at about $k=n^{1/3}$. Yamamoto proved the formula for every $k<n^{1/3-\delta}$. The site records the theorem with its range under [ErKa46].

Covers. The asymptotic count for every $k<(\log n)^{3/2-\epsilon}$. It says nothing about larger $k$, so Problem 725, which asks for an asymptotic formula without restricting $k$, is not settled by it.

Acceptance. Refereed: Amer. J. Math. 68 (1946), no. 2, 230–236; the issue is dated April 1946, and the page is dated to the first day of that month. The site's curator records the theorem under [ErKa46] while labeling the problem OPEN, which credits the partial result without settling the problem, so the page lists no reviewed evidence. The library card records the paper's theorems; the proof has not been reconstructed or independently reviewed in this corpus.