Problem Statement
Let $\hat{R}(G)$ denote the size Ramsey number, the minimal number of edges $m$ such that there is a graph $H$ with $m$ edges such that in any $2$-colouring of the edges of $H$ there is a monochromatic copy of $G$.
Let $F_1$ and $F_2$ be the union of stars. More precisely, let $F_1=\cup_{i\leq s} K_{1,n_i}$ and $F_2=\cup_{j\leq t} K_{1,m_j}$. Prove that\[\hat{R}(F_1,F_2) = \sum_{2\leq k\leq s+2}\max\{n_i+m_j-1 : i+j=k\}.\]
Let $F_1$ and $F_2$ be the union of stars. More precisely, let $F_1=\cup_{i\leq s} K_{1,n_i}$ and $F_2=\cup_{j\leq t} K_{1,m_j}$. Prove that\[\hat{R}(F_1,F_2) = \sum_{2\leq k\leq s+2}\max\{n_i+m_j-1 : i+j=k\}.\]
Categories:
Graph Theory Ramsey Theory
Progress
Burr, Erdős, Faudree, Rousseau, and Schelp [BEFRS78] proved this when all the $n_i$ are identical and all the $m_i$ are identical.See also the entry in the graphs problem collection.
Source: erdosproblems.com/561 | Last verified: January 15, 2026