PreprintCC BY 4.0arXiv
The three-colour bipartite Ramsey number of sufficiently long even cycles
Abstract
For a bipartite graph H, let br3(H) be the least integer N such that every three-colouring of KN,N contains a monochromatic copy of H. We prove that br3(C2n)=3n for all sufficiently large n, and give the lower bound for every n≥2. For the upper bound, we reduce a counterexample to a finite rational weighted profile. Cycle extension and vertex-cover arguments force the critical components to have six unique minimum covers which partition the profile. The remaining component supports give a contradiction.
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BibTeX
@misc{ai2026threecolour,
title = {{The three-colour bipartite Ramsey number of sufficiently long even cycles}},
author = {Jiangdong Ai and Yihan Yin},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.29813},
eprint = {2610.29813},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.29813},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The three-colour bipartite Ramsey number of sufficiently long even cycles
AU - Jiangdong Ai
AU - Yihan Yin
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.29813
UR - https://arxiv.org/abs/2610.29813
AB - For a bipartite graph H, let br_3(H) be the least integer N such that every three-colouring of K_N,N contains a monochromatic copy of H. We prove that br_3(C_2n)=3n for all sufficiently large n, and give the lower bound for every n≥2. For the upper bound, we reduce a counterexample to a finite rational weighted profile. Cycle extension and vertex-cover arguments force the critical components to have six unique minimum covers which partition the profile. The remaining component supports give a contradiction.
ER -