Erdős Problem #533

Let \(\delta > 0\). If \(n\) is sufficiently large and \(G\) is a graph on \(n\) vertices with no \(K_5\) and at least \(\delta n^2\) edges, must \(G\) contain a set of \(\gg_\delta n\) vertices spanning no triangle? Equivalently, writing \(\mathrm{RT}_3(n, K_5, m)\) for the maximum number of edges of a \(K_5\)-free graph on \(n\) vertices in which every triangle-free vertex set has fewer than \(m\) vertices (the *triangle Ramsey–Turán number*), is \[\delta_3(5) = \lim_{\epsilon \to 0} \lim_{n \to \infty} \frac{\mathrm{RT}_3(n, K_5, \epsilon n)}{n^2} = 0?\] This is a problem of Erdős, Hajnal, Simonovits, Sós, and Szemerédi [EHSSS94], who proved \(\delta_3(5) \leq 1/12\) and the analogous \(\delta_3(4) = 0\), and observed \(\delta_3(7) \geq 1/4\) via a construction of Erdős and Rogers [ErRo62]. The answer is **no**: Balogh and Lenz [BaLe13] disproved it by showing \(\delta_3(5) > 0\), and the exact value \(\delta_3(5) = 1/12\) was determined by the matching lower-bound construction of Liu, Reiher, Sharifzadeh, and Staden [LRSS21] (see `erdos_533.variants.lrss_lower`).