Asymptotic Tightness of the Pigeonhole Bound for Large-Order Davenport-Schinzel Sequences
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866917277832577024 |
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| author | Geneson, Jesse |
| author_facet | Geneson, Jesse |
| contents | 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 $a,b > 0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_15375 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Asymptotic Tightness of the Pigeonhole Bound for Large-Order Davenport-Schinzel Sequences Geneson, Jesse Combinatorics Discrete Mathematics 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 $a,b > 0$. |
| title | Asymptotic Tightness of the Pigeonhole Bound for Large-Order Davenport-Schinzel Sequences |
| topic | Combinatorics Discrete Mathematics |
| url | https://arxiv.org/abs/2602.15375 |