Year: 2026
Type: preprint
Abstract: We determine the size Ramsey number of a two-component star forest versus an arbitrary star forest. Let $q,a,b,n_1,\ldots,n_q$ be positive integers with $a\ge b$ and $n_1\ge\cdots\ge n_q$. We prove that \[ \begin{split} \rhat\left(K_{1,a}\sq K_{1,b}, \bigsqcup_{i=1}^{q}K_{1,n_i}\right) ={}&(a+n_1-1) +\sum_{i=1}^{q-1}\max\{a+n_{i+1}-1,b+n_i-1\}\\ &+(b+n_q-1). \end{split} \] 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.
Source: arXiv (Cornell University)
Author: You, Chunlin
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7223021948