Claim. Let $\hat r(F_1,F_2)$ be the least number of edges of a graph $G$ such that every blue-red coloring of $E(G)$ has a blue $F_1$ or a red $F_2$, and let $\mathcal C_{\mathrm{odd}}$ be the family of odd cycles. Pikhurko's Theorem 1 (p. 404) gives

for all large $n$, by an explicit construction and a greedy argument. A graph arrowing $(K_{1,n},K_3)$ arrows $(K_{1,n},\mathcal C_{\mathrm{odd}})$, and a graph is the union of a bipartite graph and a graph of maximum degree below $n$ exactly when its edges can be colored with no red odd cycle and no blue $K_{1,n}$; so a graph with at most $\binom{2n+1}2-\binom n2-1$ edges that arrows $(K_{1,n},K_3)$ refutes the statement of Problem 613 at that $n$, once edges are added to reach that count if needed, since a graph containing a graph that is not such a union is not one either (a decomposition restricts to subgraphs). The paper's remark on p. 405 records that the upper bound is strictly below the conjectured value for every $n\ge6$ and that at $n=5$ the construction with the representation $5=2+3$ has $44$ edges against the conjectured $45$; the edge count was recomputed on the problem page. The statement is therefore false for every $n\ge5$, and as a claim for every $n\ge3$ it is disproved. The cases $n=3$ and $n=4$ are not decided by the paper or by any source cited on the problem page. The paper is paged on the library's source card.

Depends on. Nothing in this wiki; the result rests on the cited paper alone. The elementary step from the arrowing form to the statement's splitting form is made on the problem page.

Acceptance. Refereed: O. Pikhurko, Size Ramsey numbers of stars versus 3-chromatic graphs, Combinatorica 21 (2001), no. 3, 403--412, received 28 May 1999 and published 1 July 2001 (Crossref), the date this page is named by. Reviewed: the site's curator, T. F. Bloom, records the problem as disproved by Pikhurko, with the bounds of Theorem 1 and the failure at $n=5$, in the problem's commentary (its key [Pi01]; page last edited 1 December 2025, accessed 2026-09-18 for the problem page), and the thread's comments of October and November 2025 point to the same theorem. The $n=5$ instance has a Lean formalization, linked above in two versions and described below; it is not listed as evidence, since nothing was built or audited in this corpus.

The Lean formalization. The file src/latest/ErdosProblems/Erdos613.lean of Boris Alexeev's repository plby/lean-proofs, linked above at its commit of 7 September 2026, for Lean and Mathlib v4.33.0, proves

not_erdos_613 : ∃ (V : Type) (G : SimpleGraph V), G.edgeSet.ncard = 44 ∧
  ∀ (color : Sym2 V → Fin 2),
    Erdos613.hasMonoStar G color 0 5 ∨ Erdos613.hasMonoTriangle G color 1

that is, a graph with $44$ edges every $2$-coloring of whose edges has a monochromatic $K_{1,5}$ in the first color or a monochromatic triangle in the second: the arrowing form of the $n=5$ counterexample above, with $44=\binom{11}2-\binom52-1$. The file names Pikhurko as informal author and Tao as formal author, so it is a formalization of this claim's $n=5$ instance and not an independent result. Its first version is the file analysis/Analysis/Misc/erdos_613.lean of the repository teorth/analysis, linked above at the commit of 4 November 2025 that added it: 1,125 lines, closing with theorem main : Pikhurko_n5_statement, the same $44$-edge arrowing statement, with no author header; the Alexeev file's header links it. It was announced in the site's thread the same day as a formalization of Pikhurko's counterexample written, by the comment's own account, with AI coding assistance; the comment names no system. The step from the arrowing form to the statement's splitting form, that such a graph is not the union of a bipartite graph and a graph of maximum degree below $5$, is the elementary coloring argument on the problem page and is not in the file; the cases $n\ge6$ are not formalized. At the linked commit the Alexeev file has 1,190 lines, no sorry, and a closing comment recording #print axioms as propext, choice and Quot.sound. The formal-conjectures statement of the problem names the file in a formal_proof attribute, and the community database lists the problem as disproved (Lean) as of its last update, dated 4 November 2025. Nothing was built or kernel-checked in this corpus and no statement-fidelity review exists, so no formalized evidence is listed.

Read depth. The conjectures, Theorem 1 and the p. 405 remarks (pp. 403--405 and 412) were read at claims-checked depth, and the edge count $44$ was recomputed from the construction; the verification of the construction (p. 405) and the proof of the lower bound (Section 3) were not checked. Nothing is independently reviewed in this corpus.