Erdős Problem #1037

Let $G$ be a graph on $n$ vertices in which every degree occurs at most twice, and the number of distinct degree is $>(\frac{1}{2}+\epsilon)n$. Must $G$ contain a trivial (empty or complete) subgraph of size 'much larger' than $\log n$?

#1037: [Er93,p.347]

graph theory

A question of Chen and Erdős.

The answer is no - Cambie, Chan, and Hunter have in the commment 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 $O(\log n)$.

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This page was last edited 22 September 2025.

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Additional thanks to: Stijn Cambie, Koishi Chan, Zach Hunter, and Mehtaab Sawhney

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