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Claim. In the notation of Problem 561, call a diagonal $k$ good if some pair $(i,j)$ with $i+j=k$ attaining $l_k$ has $n_i$ and $m_j$ both odd, or $\min(n_i,m_j)=1$. Theorem 1 ("Local criterion", p. 2) of the manuscript Local degree conditions for the size Ramsey numbers of star forests (tienxion, 4 September 2026) claims $\hat R(F_1,F_2)=\sum_{k=2}^{s+t}l_k$ whenever every diagonal $k<s+t$ that is not good satisfies $l_k-l_{k+1}\ge3$, or $l_k-l_{k+1}=2$ together with $l_{k+1}-l_{k+2}\ge2$ (when $k+1<s+t$) or $l_{s+t}\ge2$ (when $k+1=s+t$). Good diagonals and the last diagonal carry no condition, so good diagonals may repeat values; when every diagonal is good the claim is the case of every $\eta_k$ vanishing in Cipollini's deficit bound. The theorem also bounds the maximum degree of every host attaining the value. Two stated consequences: the formula holds whenever $l_k-l_{k+1}\ge2$ for all $k<s+t$ (Corollary 5), and $\hat r(K_{1,6}\sqcup rK_{1,3}\sqcup mK_2,K_{1,5}\sqcup K_2)=11+7r+5m$ for all $r,m\ge1$ (Corollary 7). By the submission, the pairs covered include ones on which the exact value exceeds the deficit lower bound by any prescribed number of edges, as Corollary 5 gives when many diagonals that are not good each drop by at least two. The proof deletes a vertex of maximum degree, bounds the maximum degrees of equality hosts by decreasing induction on $k$, and applies Fournier's theorem on edge colorings where the vertices of maximum degree induce a forest.

Submission note. Posted to erdosproblems.com as a proof claim by tienxion (account tienxion) on 4 September 2026, giving "OpenAI Codex (GPT-6), including parallel research agents" as the AI used:

We prove the conjectured size Ramsey formula for pairs of star forests whose diagonal maxima satisfy explicit local gap conditions. The proof builds on earlier simultaneous-tail lower bound by controlling the maximum-degree vertices in equality cases and applying Fournier’s edge-colouring theorem. This gives exact values for additional families, including families where it improves the earlier lower bound by arbitrarily many edges, but does not fully resolve this problem. Notes: This is a partial result. The simultaneous-tail framework, good-diagonal definition, and parity-dependent deficit bound are due to Ricky Cipollini (rickyc), whose 6 August 2026 contribution is explicitly cited in the manuscript. The additional step is equality-host degree control under local gap conditions. AI assistance was used in developing and writing the draft; the submitting human reports mathematical review and supplied the summary. External peer review has not taken place, and priority of the additional criterion remains to be confirmed. The linked PDF is the reviewed version 1.

Covers. The formula for every pair of star forests in which each diagonal before the last whose maximum is attained by no pair of odd sizes and no pair containing a single-edge star is followed by a drop of at least three, or by a drop of exactly two and then a further drop of at least two (or, when the next diagonal is the last, a last value of at least two); the formula for all star forests is not claimed.

Depends on. Nothing in this wiki; the manuscript reproves the deletion lemma it credits to Cipollini's comment of 6 August 2026 (its Lemma 2) and the parity coloring of Davoodi, Javadi, Kamranian and Raeisi, Lemma 2.1 (its Lemma 3), and cites Vizing's theorem and Fournier's edge-coloring theorem. The manuscript credits the deletion framework, the distinction between good and other diagonals and the deficit bound to Cipollini and says that it does not claim these ingredients as new.

Standing. Claimed. The tab names OpenAI Codex (GPT-6), including parallel research agents, as the AI system used; the submitter's notes and the manuscript's draft-status note say that AI assistance was used in developing and writing the manuscript, that the submitter reviewed the mathematics, that no outside peer review has taken place and that the priority of the added criterion remains to be confirmed. The claim page states the theorem from the manuscript; its proof was not checked, so no evidence kind is listed. The claim had no comments on the tab (2026-10-06); the site's label is OPEN and its commentary does not mention the claim (page last edited 1 February 2026); no arXiv version, journal record or independent review was found.