Erdős Problem #1033

Let \(h(n)\) be such that every graph on \(n\) vertices with \(>n^2/4\) many edges contains a triangle whose vertices have degrees summing to at least \(h(n)\). Estimate \(h(n)\). In particular, is it true that \[h(n)\geq (2(\sqrt{3}-1)-o(1))n?\] A conjecture of Bollobás and Erdős. The answer is no. A blow-up of the butterfly graph, posted on the [Erdős problems forum](https://www.erdosproblems.com/forum/thread/1033) in June and July 2026 and written up in [this note](https://woutercvb.github.io/Erdos1033_smalloptimization.pdf), gives arbitrarily large graphs with more than \(n^2/4\) edges in which every triangle has degree sum at most \(1.464n\), which is less than \(2(\sqrt{3}-1)n\). Estimating the optimal value of \(h(n)\) is still open.