Size Ramsey numbers of star forests with two components on one side

We determine the size Ramsey number of a two-component star forest versus an arbitrary star forest. Let $q,a,b,n_1,\dots ,n_q$ be positive integers with $a\ge b$ and $n_1\ge \cdots \ge n_q$. We prove that $$\begin{array}{rl}{\displaystyle \text{rhat}(K_{1,a}\text{sq}{K}_{1,b},\underset{i=1}{\overset{q}{⨆}}{K}_{1,n_i})=}& {\displaystyle (a+n_1-1)+\sum _{i=1}^{q-1}\max \{a+n_{i+1}-1,b+n_i-1\}}\\ {\displaystyle }& {\displaystyle +(b+n_q-1)\mathrm{.}}\end{array}$$ 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.