Asymptotic Tightness of the Pigeonhole Bound for Large-Order Davenport-Schinzel Sequences
🤖 AI Summary
This study investigates the asymptotic tightness of the classical pigeonhole upper bound λ(s, m) for higher-order Davenport–Schinzel sequences in the regime where s/√m → ∞. By combining combinatorial structural analysis with asymptotic methods, the authors establish for the first time that λ(n, n)/n³ → 1/2 as n → ∞, and further generalize this to arbitrary proportional scaling, proving λ(an, bn) ∼ (ab²/2)n³ for fixed positive constants a and b. These results precisely determine the leading constant that had remained unresolved in prior work. Consequently, the classical upper bound λ(s, m) ≤ C(m, 2)(s + 1) is shown to be asymptotically tight when s/√m tends to infinity, thereby resolving a long-standing uncertainty regarding the sharpness of this bound in this parameter regime.
Application Category
📝 Abstract
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 $a,b > 0$.
Problem
Research questions and friction points this paper is trying to address.
Davenport-Schinzel sequences
asymptotic tightness
pigeonhole bound
combinatorial geometry
extremal combinatorics
Innovation
Methods, ideas, or system contributions that make the work stand out.
Davenport-Schinzel sequences
asymptotic tightness
pigeonhole bound
combinatorial geometry
extremal combinatorics
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