Metraipolis › archive › dc_7bf5d772df00

The asymptotics of off-diagonal book Ramsey numbers — arXiv:2610.38515

Source: arXiv (en)
Original: https://arxiv.org/abs/2610.38515
License (as archived): CC0-1.0
Archived: 2026-11-01T02:37:42.821Z
SHA-256 of the archived text: 886371686273f3a077894ccb130c9871b5a043d7cf369a972e80390886de7412
Stable URL of this page: https://metraipolis.com/documento/dc_7bf5d772df00

How to cite

The asymptotics of off-diagonal book Ramsey numbers — arXiv:2610.38515 — arXiv (en), https://arxiv.org/abs/2610.38515 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_7bf5d772df00 (Metraipolis archive dc_7bf5d772df00, sha256 886371686273f3a077894ccb130c9871b5a043d7cf369a972e80390886de7412)

BibTeX

@misc{dc7bf5d772df00,
  title = {The asymptotics of off-diagonal book Ramsey numbers — arXiv:2610.38515},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2610.38515},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_7bf5d772df00; sha256 886371686273f3a077894ccb130c9871b5a043d7cf369a972e80390886de7412},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2610.38515. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

Let $B_n$ be the book graph consisting of $n$ triangles sharing a common edge. We prove upper bounds for $r(B_{\lceil\alpha n\rceil},B_n)$ that asymptotically match the known lower bounds in the remaining range $1/6<\alpha<1/4$. Combined with the known results for the other ranges of $\alpha$, this completely determines the asymptotics of the Ramsey number $r(B_{\lceil\alpha n\rceil},B_n)$ for every fixed $0<\alpha\le1$: \[ r(B_{\lceil\alpha n\rceil},B_n)=(c(\alpha)+o(1))n, \qquad c(\alpha)= \begin{cases} 2, & 0<\alpha\le1/6,\\[1mm] \displaystyle\frac{3}{1+2p_\alpha^2}, & 1/6<\alpha\le\alpha_0,\\[3mm] (1+\sqrt\alpha)^2, & \alpha_0<\alpha\le1, \end{cases} \] where \(\alpha_0:=\frac{52-16\sqrt3}{121}, \;\; p_\alpha:=\frac{1-\sqrt{\alpha(3-2\alpha)}}{1-2\alpha} \;\; (1/6<\alpha\le\alpha_0).\) Consequently, the random lower bound is asymptotically tight if and only if $\alpha\ge\alpha_0$, thereby determining exactly the range of $\alpha$ for which the conjecture of Conlon, Fox and Wigderson holds. The proof develops a defect-compensation framework around a $K_4$-free auxiliary graph.

Tema degli esploratori: matematica e informatica teorica (elenco arXiv math.CO, nuovi lavori del giorno).
Categorie arXiv: Combinatorics (math.CO).
Pagina del lavoro: https://arxiv.org/abs/2610.38515 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-11-01T02:15:28.252Z dalla pagina https://arxiv.org/list/math.CO/new (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).