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PreprintarXiv licencearXiv

The asymptotics of off-diagonal book Ramsey numbers

  • Lulu Dai
  • Chunchao Fan
  • Qizhong Lin
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Abstract

Let Bn be the book graph consisting of n triangles sharing a common edge. We prove upper bounds for r(B⌈αn⌉,Bn) that asymptotically match the known lower bounds in the remaining range 1/6<α<1/4. Combined with the known results for the other ranges of α, this completely determines the asymptotics of the Ramsey number r(B⌈αn⌉,Bn) for every fixed 0<α≤1: r(B⌈αn⌉,Bn)=(c(α)+o(1))n, c(α)= 2, & 0<α≤1/6, [1mm] 3/(1+2pα2), & 1/6<α≤α0, [3mm] (1+√α)2, & α0<α≤1, where α0:=(52−16√3)/121, pα:=(1−√(α(3−2α)))/(1−2α) (1/6<α≤α0). Consequently, the random lower bound is asymptotically tight if and only if α≥α0, thereby determining exactly the range of α for which the conjecture of Conlon, Fox and Wigderson holds. The proof develops a defect-compensation framework around a K4-free auxiliary graph.

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Cite

BibTeXRIS
BibTeX
@misc{dai2026asymptotics,
  title = {{The asymptotics of off-diagonal book Ramsey numbers}},
  author = {Lulu Dai and Chunchao Fan and Qizhong Lin},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.38515},
  eprint = {2610.38515},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.38515},
}
RIS
TY  - UNPB
TI  - The asymptotics of off-diagonal book Ramsey numbers
AU  - Lulu Dai
AU  - Chunchao Fan
AU  - Qizhong Lin
PY  - 2026
DA  - 2026/10/28
PB  - arXiv
DO  - 10.48550/arxiv.2610.38515
UR  - https://arxiv.org/abs/2610.38515
AB  - Let B_n be the book graph consisting of n triangles sharing a common edge. We prove upper bounds for r(B_⌈αn⌉,B_n) that asymptotically match the known lower bounds in the remaining range 1/6<α<1/4. Combined with the known results for the other ranges of α, this completely determines the asymptotics of the Ramsey number r(B_⌈αn⌉,B_n) for every fixed 0<α≤1: r(B_⌈αn⌉,B_n)=(c(α)+o(1))n, c(α)= 2, & 0<α≤1/6, [1mm] 3/(1+2p_α^2), & 1/6<α≤α_0, [3mm] (1+√α)^2, & α_0<α≤1, where α_0:=(52−16√3)/121, p_α:=(1−√(α(3−2α)))/(1−2α) (1/6<α≤α_0). Consequently, the random lower bound is asymptotically tight if and only if α≥α_0, thereby determining exactly the range of α for which the conjecture of Conlon, Fox and Wigderson holds. The proof develops a defect-compensation framework around a K_4-free auxiliary graph.
ER  -