Erdős Problem #619

The problem

For a triangle-free graph $G$ let $h_r(G)$ be the smallest number of edges that need to be added to $G$ so that it has diameter $r$ (while preserving the property of being triangle-free).

Is it true that there exists a constant $c>0$ such that if $G$ is a connected graph on $n$ vertices then $h_4(G)<(1-c)n$?

Statement quoted from erdosproblems.com/619, maintained by Thomas Bloom. Original source: [EGR98,p.499][Er99,p.4]. Current status there: solved (Lean).

AI contributions

As recorded in Terence Tao's wiki AI contributions to Erdős problems (data as of 30 Jun 2026). 🟢 full, 🟡 partial, ⚪ unverified, 🔴 incorrect.

Caveats

The wiki's classifications are provisional and are not a benchmark. Many Erdős problems have never had a thorough literature review, so a result that looks new may later turn out to be known; some problems were also stated ambiguously or incorrectly. See the problem page for the full discussion.

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