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

TL;DR

This paper establishes the asymptotic tightness of the pigeonhole bound for large-order Davenport-Schinzel sequences, providing precise growth rates and resolving longstanding constants in the sequence complexity analysis.

Contribution

It proves the asymptotic tightness of the pigeonhole upper bound for large-order Davenport-Schinzel sequences and determines exact growth constants.

Findings

inom{m}{2}(s+1) is asymptotically tight for large s/m

 extlambda(n,n)/n^3 o 1/2, resolving previous bounds

 extlambda(an,bn) o (ab^2/2) n^3 for constants a,b > 0

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 $a,b > 0$.