We prove that the pigeonhole upper bound $λ(s,m) \leq \binom{m}{2}(s+1)$ is asymptotically tight whenever $s/\!\sqrt{m} \to \infty$. In particular, $λ(s,m) \sim \binom{m}{2}\,s$ in this regime. As corollaries: $λ(n,n)/n^3 \to \frac{1}{2}$, resolving the leading constant from the previously known interval $[\frac{1}{3}, \frac{1}{2}]$; and more generally $λ(an,bn) \sim \frac{ab^2}{2}\,n^3$ for any constants $a,b > 0$.

翻译：我们证明，当 $s/\!\sqrt{m} \to \infty$ 时，鸽巢上界 $λ(s,m) \leq \binom{m}{2}(s+1)$ 是渐近紧的。特别地，在此条件下有 $λ(s,m) \sim \binom{m}{2}\,s$。作为推论：$λ(n,n)/n^3 \to \frac{1}{2}$，这确定了先前已知区间 $[\frac{1}{3}, \frac{1}{2}]$ 中的主导常数；更一般地，对于任意常数 $a,b > 0$，有 $λ(an,bn) \sim \frac{ab^2}{2}\,n^3$。