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

Jesse Geneson

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 $a,b > 0$.

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

Abstract

We prove that the pigeonhole upper bound is asymptotically tight whenever . In particular, in this regime. As corollaries: , resolving the leading constant from the previously known interval ; and more generally for any constants .
Paper Structure (8 sections, 6 theorems, 12 equations)

This paper contains 8 sections, 6 theorems, 12 equations.

Table of Contents

  1. Introduction
  2. A Uniform Lower Bound via the Roselle--Stanton Construction
  3. A Uniform Lower Bound
  4. The $S_2$ Construction
  5. Sequence
  6. Properties
  7. Main Result
  8. Corollaries

Key Result

Lemma 2

$|\mathrm{Alt}(s,m)| = 2(m-1)\lceil(s-2)/2\rceil + 1$.

Theorems & Definitions (14)

  • Definition 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Theorem 4
  • proof
  • Corollary 5
  • proof
  • Corollary 6
  • ...and 4 more