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Star-critical Ramsey numbers involving large generalized books
Abstract
For graphs F, G, and H, we write F → (G, H) if every red/blue edge-coloring of F contains either a red copy of G or a blue copy of H. The Ramsey number $R(G, H)$ is the smallest integer N such that KN → (G, H). Let H := Kk + nKh be the generalized book graph, and let G := Kp+1(a1, a2, …, ap+1) be a complete (p+1)-partite graph satisfying a1 = 1, a2 | (nh−1), and ai ≤ ai+1. In this paper, avoiding the use of Szemerédi's regularity lemma, we prove that for any fixed h, p ≥ 1, k ≥ 2, and sufficiently large n, Kp(nh + a2k − 1) + 1 ∖ K1, nh − a2(h−2) − 1→(G, H). This result yields the star-critical Ramsey number r*(G, H) = (p−1)(nh + a2k − 1) + a2(k + h − 2) + 1.
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BibTeX
@misc{huang2026starcritical,
title = {{Star-critical Ramsey numbers involving large generalized books}},
author = {Peiyu Huang and Qizhong Lin and Lin Niu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.11151},
eprint = {2610.11151},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.11151},
}
RIS
TY - UNPB
TI - Star-critical Ramsey numbers involving large generalized books
AU - Peiyu Huang
AU - Qizhong Lin
AU - Lin Niu
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.11151
UR - https://arxiv.org/abs/2610.11151
AB - For graphs F, G, and H, we write F → (G, H) if every red/blue edge-coloring of F contains either a red copy of G or a blue copy of H. The Ramsey number $R(G, H)$ is the smallest integer N such that K_N → (G, H). Let H := K_k + nK_h be the generalized book graph, and let G := K_p+1(a_1, a_2, …, a_p+1) be a complete (p+1)-partite graph satisfying a_1 = 1, a_2 | (nh−1), and a_i ≤ a_i+1. In this paper, avoiding the use of Szemerédi's regularity lemma, we prove that for any fixed h, p ≥ 1, k ≥ 2, and sufficiently large n, K_p(nh + a_2k − 1) + 1 ∖ K_1, nh − a_2(h−2) − 1→(G, H). This result yields the star-critical Ramsey number r_*(G, H) = (p−1)(nh + a_2k − 1) + a_2(k + h − 2) + 1.
ER -