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Book Ramsey numbers via algebraic constructions

  • Lulu Dai
  • Qizhong Lin
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Abstract

Let Bn denote the book graph consisting of n triangles sharing a common edge. Rousseau and Sheehan (1978) proved, using Paley graphs, that R(Bn,Bn)=4n+2 whenever 4n+1 is a prime power. Since then, only a few additional exact values of R(Bn,Bn) have been obtained. In this paper, we obtain R(Bn,Bn)=4n+1 for infinitely many n by constructing graphs from pseudo-cyclic strongly regular graphs (PC-graphs). To the best of our knowledge, this is the first new infinite family of exact diagonal book Ramsey numbers since 1978. We also prove that R(Bn−2,Bn)≤ 4n−3 for every n≥ 3 with n≠ 6, removing the original condition n≡ 2pmod 3 imposed by Rousseau and Sheehan. Moreover, equality R(Bn−2,Bn)=4n−3 holds if there exists a symmetric Hadamard matrix of order 2n−2 with all diagonal entries equal to 1. In particular, this equality holds for every n=22ℓ−1+1 with ℓ≥ 1.

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Cite

BibTeXRIS
BibTeX
@misc{dai2026book,
  title = {{Book Ramsey numbers via algebraic constructions}},
  author = {Lulu Dai and Qizhong Lin},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2606.07214},
  eprint = {2606.07214},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.07214},
}
RIS
TY  - UNPB
TI  - Book Ramsey numbers via algebraic constructions
AU  - Lulu Dai
AU  - Qizhong Lin
PY  - 2026
DA  - 2026/10/13
PB  - arXiv
DO  - 10.48550/arxiv.2606.07214
UR  - https://arxiv.org/abs/2606.07214
AB  - Let B_n denote the book graph consisting of n triangles sharing a common edge. Rousseau and Sheehan (1978) proved, using Paley graphs, that R(B_n,B_n)=4n+2 whenever 4n+1 is a prime power. Since then, only a few additional exact values of R(B_n,B_n) have been obtained. In this paper, we obtain R(B_n,B_n)=4n+1 for infinitely many n by constructing graphs from pseudo-cyclic strongly regular graphs (PC-graphs). To the best of our knowledge, this is the first new infinite family of exact diagonal book Ramsey numbers since 1978. We also prove that R(B_n−2,B_n)≤ 4n−3 for every n≥ 3 with n≠ 6, removing the original condition n≡ 2pmod 3 imposed by Rousseau and Sheehan. Moreover, equality R(B_n−2,B_n)=4n−3 holds if there exists a symmetric Hadamard matrix of order 2n−2 with all diagonal entries equal to 1. In particular, this equality holds for every n=2^2ℓ−1+1 with ℓ≥ 1.
ER  -