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Size Ramsey numbers of star forests with two components on one side
Abstract
We determine the size Ramsey number of a two-component star forest versus an arbitrary star forest. Let q,a,b,n1,…,nq be positive integers with a≥ b and n1≥⋯≥ nq. We prove that rhat(K1,asq K1,b, bigsqcupi=1qK1,ni) =&(a+n1−1) +∑i=1q−1max{a+ni+1−1,b+ni−1} &+(b+nq−1). This proves the conjecture of Burr, Erdős, Faudree, Rousseau and Schelp (Erdős Problem~\#561) in the case where one of the two star forests has exactly two components. The formula holds without any restriction on the parity of the star sizes. The proof combines vertex deletion, iterated matching deletion, and Lovász's factor theorem.
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BibTeX
@misc{you2026size,
title = {{Size Ramsey numbers of star forests with two components on one side}},
author = {Chunlin You},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.25958},
eprint = {2610.25958},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.25958},
}
RIS
TY - UNPB
TI - Size Ramsey numbers of star forests with two components on one side
AU - Chunlin You
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.25958
UR - https://arxiv.org/abs/2610.25958
AB - We determine the size Ramsey number of a two-component star forest versus an arbitrary star forest. Let q,a,b,n_1,…,n_q be positive integers with a≥ b and n_1≥⋯≥ n_q. We prove that rhat(K_1,asq K_1,b, bigsqcup_i=1^qK_1,n_i) =&(a+n_1−1) +∑_i=1^q−1max{a+n_i+1−1,b+n_i−1} &+(b+n_q−1). This proves the conjecture of Burr, Erdős, Faudree, Rousseau and Schelp (Erdős Problem~\#561) in the case where one of the two star forests has exactly two components. The formula holds without any restriction on the parity of the star sizes. The proof combines vertex deletion, iterated matching deletion, and Lovász's factor theorem.
ER -