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Multicolor Ramsey numbers of unbalanced double stars
Abstract
For integers n≥ m≥1, let S(n,m) be the double star obtained by joining the centers of K1,n and K1,m. Ruotolo and Song proved the upper bound r(S(n,m);k)≤ kn+m+2 under the condition (n+1)⌈(n+1)/(k−1)⌉ >m((k−1)n+m), as well as the matching lower bound for odd k. Zhao, Mao and Zhou proved the matching lower bound for even k and even n. We establish the remaining lower bound for even k≥6 and odd n through a new edge-coloring construction. Combining these results, we obtain r(S(n,m);k)=kn+m+2 for all integers k≥5 and n≥ m≥1 satisfying (n+1)⌈(n+1)/(k−1)⌉ >m((k−1)n+m). We also determine the four-color Ramsey number for m≥2 and odd n≥15m, showing that r(S(n,m);4)=4n+m+1.
Related
BibTeX
@misc{niu2026multicolor,
title = {{Multicolor Ramsey numbers of unbalanced double stars}},
author = {Lin Niu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.24523},
eprint = {2610.24523},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.24523},
}
RIS
TY - UNPB
TI - Multicolor Ramsey numbers of unbalanced double stars
AU - Lin Niu
PY - 2026
DA - 2026/10/17
PB - arXiv
DO - 10.48550/arxiv.2610.24523
UR - https://arxiv.org/abs/2610.24523
AB - For integers n≥ m≥1, let S(n,m) be the double star obtained by joining the centers of K_1,n and K_1,m. Ruotolo and Song proved the upper bound r(S(n,m);k)≤ kn+m+2 under the condition (n+1)⌈(n+1)/(k−1)⌉ >m((k−1)n+m), as well as the matching lower bound for odd k. Zhao, Mao and Zhou proved the matching lower bound for even k and even n. We establish the remaining lower bound for even k≥6 and odd n through a new edge-coloring construction. Combining these results, we obtain r(S(n,m);k)=kn+m+2 for all integers k≥5 and n≥ m≥1 satisfying (n+1)⌈(n+1)/(k−1)⌉ >m((k−1)n+m). We also determine the four-color Ramsey number for m≥2 and odd n≥15m, showing that r(S(n,m);4)=4n+m+1.
ER -