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Claim. In the notation of Problem 561, the paper proves the conjectured formula $\hat r(F_1,F_2)=\sum_{k=2}^{s+t}l_k$ in four families, numbered as in the journal version:

• Theorem 2.3: for $s=1$, "For given positive integers $n$ and $m_1\geq m_2\geq\cdots\geq m_t\geq2$, we have $\hat{r}(K_{1,n},\bigsqcup_{j=1}^tK_{1,m_j})=\sum_{j=1}^t(n+m_j-1)$", with the extremal graphs (library page).
• Theorem 2.4: for $F_1=2K_{1,n}$ and $m_t\ge2$, $\hat r(F_1,F_2)=n+m_1-1+\sum_{i=1}^t(n+m_i-1)$ (library page).
• Theorem 2.5: the formula whenever all $n_i$ and all $m_j$ are odd, single-edge stars included (library page).
• Theorem 2.6: for $F_1=sK_{1,n}$ with $n$ and $m_1$ odd and $m_t\ge2$, $\hat r(F_1,F_2)=(s-1)(n+m_1-1)+\sum_{j=1}^t(n+m_j-1)$, with the extremal graph (library page).

In each family the stated value is the sum of the diagonal maxima. The mechanism is the paper's Lemma 2.1 (p. 3): a graph with $\Delta(G)\le m+n-3$, or with $\Delta(G)\le m+n-2$ when $m$ and $n$ are both odd, has a red-blue coloring with no red $K_{1,n}$ and no blue $K_{1,m}$, so an arrowing graph has a vertex of large degree, which is deleted and the argument repeated. Theorem 2.2 reproves the uniform case of Burr, Erdős, Faudree, Rousseau and Schelp 1978 with a shorter argument and completes its list of extremal graphs. The paper states the general formula as open and extends it to $q$ colors as its Conjecture 3.1 (p. 9). The theorems are paged on the library's source card.

Covers. The formula for $s=1$ with $m_t\ge2$; for $s=2$, $n_1=n_2$ and $m_t\ge2$; for all $n_i$ and $m_j$ odd; and for all $n_i$ equal to one odd $n$ with $m_1$ odd and $m_t\ge2$. The formula for all star forests is not claimed.

Depends on. Nothing in this wiki; the result rests on the cited paper, whose Lemma 2.1 uses Vizing's theorem and Petersen's $2$-factorization theorem.

Acceptance. Refereed: A. Davoodi, R. Javadi, A. Kamranian and G. Raeisi, On a conjecture of Erdős on size Ramsey number of star forests, Ars Math. Contemp. 25 (2025), no. 2, #P2.09, 10 pp. (received 4 May 2023, accepted 10 May 2024, published online 1 April 2025). The page is dated by the first posting, arXiv:2111.02065 (3 November 2021), by the same four authors under the same title. The site's commentary credits the further special cases to this paper, but the site labels the problem OPEN, so its pages are not acceptance.

Read depth. The statements of Theorems 2.2--2.6 and Lemma 2.1 were read in the journal version; the proofs were read for structure only, and the arXiv versions were not compared with it. Nothing is independently reviewed in this corpus.