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Claim. The statement of Problem 533 is false: there is no function $c(\delta)>0$ such that every $K_5$-free graph on $n$ vertices with at least $\delta n^2$ edges has, for large $n$, a set of $c(\delta)n$ vertices spanning no triangle. Balogh and Lenz prove in Theorem 3 that for $t\ge2$ and $2\le\ell\le t$, with $u=\lceil t/2\rceil$, $\theta_t(K_{t+\ell})\ge\frac12(1-\frac1\ell)2^{-u^2}$, where $\theta_t(H)$ is the limit of $\mathrm{RT}_t(n,H,\epsilon n)/n^2$ as $n\to\infty$ and then $\epsilon\to0$. At $t=3$, $\ell=2$ this is $\theta_3(K_5)\ge1/64$, the site's $\delta_3(5)\ge1/64>0$: for every $\epsilon>0$ and all large $n$ there is a $K_5$-free graph on $n$ vertices with $\alpha_3\le\epsilon n$ and at least $(1/64-o(1))n^2$ edges. With $\delta=1/128$ no $c(\delta)$ exists; the deduction is written out on the problem page, together with the normalization $\theta_3(K_5)=\delta_3(5)$. The paper poses the question as its Problem 2 and calls the answer its main result; the constructions come from a hypergraph statement built with high-dimensional sphere geometry. Liu, Reiher, Sharifzadeh and Staden later fixed the exact threshold $\delta_3(5)=1/12$, recorded on their claim page.

Acceptance. Refereed: Israel Journal of Mathematics 194 (2013), no. 1, 45--68, doi:10.1007/s11856-012-0076-2 (published online 29 June 2012; the Crossref record and the arXiv listing's journal reference of 2026-09-18). The text cited is the arXiv v2 of 22 September 2011; the journal text was not compared. Reviewed: the site's curator, Thomas F. Bloom, credits the disproof to Balogh and Lenz in the problem's commentary (page labeled DISPROVED (LEAN), last edited 27 January 2026, accessed 2026-09-18), and the curator's thread comment of 27 January 2026 says that $\delta>0$ appears to have been proved earlier by Balogh and Lenz; the proof-claim tab is empty. Not formalized: the Lean file Erdos533.lean in plby/lean-proofs, which formal-conjectures names as the formal proof of erdos_533 (; absent from the commit at main on 2026-09-18), lists Balogh and Lenz among its informal authors, with Codex and GPT-5.6 Sol as formal authors, but builds the Liu--Reiher--Sharifzadeh--Staden construction and not Theorem 3, so it is linked from their claim page; the corpus has not built it. Proof coverage is statements only: Theorem 3, Corollary 4 and the p. 4 displays were checked, and no proof was read. The claim rests on the source card balogh_2013_ramsey_turan_numbers_graphs_hypergraphs and consumes no page of this wiki.