Asymptotic Tightness of the Pigeonhole Bound for Large-Order Davenport-Schinzel Sequences

Abstract

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