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Claim. József Balogh, Felix Christian Clemen and Bernard Lidický, Max Cuts in Triangle-free Graphs, arXiv:2103.14179v1, submitted 25 March 2021 (the claim's date), prove as Theorem 2(b),(c), p. 2, that there is an $N_0$ such that every triangle-free graph $G$ on $N\ge N_0$ vertices satisfies $D_2(G)\le N^2/25$ when $|E(G)|\ge0.3197\binom{N}{2}$ or $|E(G)|\le0.2486\binom{N}{2}$. Here $D_2(G)$ is the fewest edges whose deletion makes $G$ bipartite; the paper writes $n$ for the order and gives no value of $N_0$. With $N=5n$ this answers the question of Problem 23 yes, at most $n^2$ deletions, for every sufficiently large $n$ and every triangle-free graph on $5n$ vertices with at least $0.3197\binom{5n}{2}$ or at most $0.2486\binom{5n}{2}$ edges. The statement and this specialization are recorded on the Theorem 2 page of the source card. The same six-page extended abstract appeared in Extended Abstracts EuroComb 2021 (Trends in Mathematics 14, Birkhäuser), pp. 509--514, online 24 August 2021.

Covers. The question of Problem 23 for the triangle-free graphs on $5n$ vertices with at least $0.3197\binom{5n}{2}$ or at most $0.2486\binom{5n}{2}$ edges, for every $n$ above an unspecified threshold. The thresholds are fractions of $\binom{5n}{2}$, not of $(5n)^2$. Not covered: the graphs whose edge count lies strictly between the two thresholds, at every order, and every order below the threshold, so neither a proof for all $n$ nor a counterexample follows. The balanced blow-up of $C_5$ (five independent sets of $n$ vertices, complete bipartite graphs between cyclically consecutive sets) has $5n^2$ edges, in the dense range, and needs at least $n^2$ deletions (p. 1), so the bound cannot be lowered there. The theorem's clause (a), $D_2(G)\le N^2/23.5$ for every triangle-free graph on $N\ge N_0$ vertices, is $(50/47)n^2$ on $5n$ vertices; it answers the question for no graph, since $50/47>1$, and the problem page records it among the known results.

Relation to other claims. Ferudun's pending claim uses clauses (b) and (c) for its two density tails: its Corollary 4.2 states them at every order, and its Theorem 1.1 uses them up to order $200$, adding a middle-band certificate of its own to assert the bound $n^2$ at every density for $1\le n\le40$. Plasma AI's pending claim carries clauses (b) and (c) to every order by the blow-up identity $D_2(G[t])=t^2D_2(G)$, with the densities measured as $2e(G)/N^2$ rather than as fractions of $\binom{N}{2}$, and clause (a) as $D_2(G)\le2N^2/47$. Neither page is a premise of this one.

Depends on. No page of this wiki; the result is the paper's own theorem, recorded on the Theorem 2 page.

Standing. Claimed, not accepted. The result's only publication is the six-page extended abstract: arXiv:2103.14179v1, the only version, which carries no journal reference, and the EuroComb 2021 chapter, DOI 10.1007/978-3-030-83823-2_82. The volume's preface says its program committee selected 135 of the 177 submitted contributions for presentation; it describes no refereeing, and an extended abstract in a proceedings volume is not refereed evidence. The abstract itself gives only a proof sketch. Its flag-algebra argument, Section 2.1, pp. 3--4, rests on a semidefinite program solved by computer, whose certificate the abstract does not supply. Section 2.2, pp. 4--5, treats densities slightly below $2/5$ through cited structural theorems of Erdős, Győri and Simonovits and of Häggkvist, and omits its detailed computations. No journal version or later full proof was found in a search on 2026-10-18; the authors' 2022 paper 10 Problems for Partitions of Triangle-free Graphs (arXiv:2203.15764) still cites the EuroComb abstract for this work. The erdosproblems.com commentary (read 2026-10-18) credits [BCL21] only with the general bound of at most $1.064n^2$ deletions, clause (a), on a problem Bloom's catalog labeled FALSIFIABLE; that is commentary on an open problem, not reviewed evidence. The source card's reading checked the statement against the PDF and read the sketch for structure; no reconstruction, certificate replay or independent review is recorded here. A refereed full proof, or an independent reviewer's acceptance of a reconstructed proof, would move the claim to accepted. The claim is partial, so the problem's standing is unchanged by it.