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Claim. Every subdivision of $K_4$ on at least six vertices is Ramsey size-linear: for each such fixed graph $H$, $R(H,F)=O(e(F))$ for every graph $F$ with no isolated vertices. This is Theorem 4 of the paper, printed p. 227 of the SIAM edition and p. 2 of arXiv:2202.10388v2. It answers the corrected Statement of Problem 566 yes for these graphs.

Covers. The corrected Statement (every subgraph on $k\ge2$ vertices has at most $2k-3$ edges) for every $G$ that is a subdivision of $K_4$ with at least six vertices. Each such graph meets the hypothesis (elementary checks made here): it has $e=v+2$, so a subgraph on $k$ vertices has at most $k+2$ edges, which is at most $2k-3$ for $k\ge5$; a $K_4$ inside it would have to sit on the four branch vertices, at least one of whose six edges is subdivided, so a subgraph on four vertices has at most $5=2\cdot4-3$ edges; and on two and three vertices the bound holds in every graph. The five-vertex subdivision $K_4^*$, which also meets the hypothesis, is outside the theorem and undecided (Problem 567).

Depends on. Nothing in this wiki; the result rests on the cited paper alone.

Acceptance. Refereed: SIAM J. Discrete Math. 38 (2024), no. 1, 225--242, received 1 March 2022, accepted in revised form 10 August 2023 and published electronically 9 January 2024. The page is dated by the first arXiv posting, 21 February 2022. The site labels the problem OPEN, and its commentary on Problem 566 does not mention the paper.

Read depth. The statement of Theorem 4 and the paper's definition of Ramsey size-linearity were checked in both editions; the proof (Section 4) was not checked. Nothing is independently reviewed in this corpus.