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PreprintCC BY 4.0arXiv

An 8+9 gluing computation for the Ramsey number R(F3,F4)

  • Hanxin Jiang
  • Zehui Shao
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Abstract

Let Ft be the graph consisting of t triangles with a common vertex and otherwise disjoint vertices. We prove that R(F3,F4)=17 using an 8+9 gluing computation. Neighbourhood matching arguments give degree and clique restrictions that reduce the search on seventeen vertices to $88,294,674$ ordered seed pairs. The verified main computation excludes $88,294,634$ pairs. A cross-edge counting argument excludes thirty-four of the forty exceptional pairs, and independently checked finite-domain certificates exclude the remaining six. Together with the construction whose red graph is the complete bipartite graph K8,8, these results establish R(F3,F4)=17.

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Cite

BibTeXRIS
BibTeX
@misc{jiang2026gluing,
  title = {{An $8+9$ gluing computation for the Ramsey number $R(F_3,F_4)$}},
  author = {Hanxin Jiang and Zehui Shao},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.26109},
  eprint = {2610.26109},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.26109},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - An 8+9 gluing computation for the Ramsey number R(F_3,F_4)
AU  - Hanxin Jiang
AU  - Zehui Shao
PY  - 2026
DA  - 2026/10/19
PB  - arXiv
DO  - 10.48550/arxiv.2610.26109
UR  - https://arxiv.org/abs/2610.26109
AB  - Let F_t be the graph consisting of t triangles with a common vertex and otherwise disjoint vertices. We prove that R(F_3,F_4)=17 using an 8+9 gluing computation. Neighbourhood matching arguments give degree and clique restrictions that reduce the search on seventeen vertices to $88,294,674$ ordered seed pairs. The verified main computation excludes $88,294,634$ pairs. A cross-edge counting argument excludes thirty-four of the forty exceptional pairs, and independently checked finite-domain certificates exclude the remaining six. Together with the construction whose red graph is the complete bipartite graph K_8,8, these results establish R(F_3,F_4)=17.
ER  -