The board / graph theory / #1037
Erdős Problem #1037
DISPROVED LEANdatabase updated 2026-01-18
The statement as posed is false. A solution has also been checked by machine, in Lean.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
A set `s` of vertices of a graph `G` is *trivial* if the subgraph it induces is empty or complete, that is, if `s` is an independent set or a clique of `G`.
Let \(G\) be a graph on \(n\) vertices in which every degree occurs at most twice, and the number of distinct degrees is \(>(\frac{1}{2}+\epsilon)n\). Must \(G\) contain a trivial (empty or complete) subgraph of size 'much larger' than \(\log n\)? A question of Chen and Erdős. The answer is no - Cambie, Chan, and Hunter have in the comment section given a simple construction of a graph on \(n\) vertices with at least \(\frac{3}{4}n\) distinct degrees, every degree appears at most twice, and the largest trivial subgraph has size \(O(\log n)\).
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved · 1 further variant / known partial result in the Lean file. The authoritative wording is on erdosproblems.com/1037.Prizenone
Topicsgraph theory
Statement in LeanFormal Conjectures
Solution in Leanyes
Original pageerdosproblems.com/1037
Attempts
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