Let $G$ be a graph with $n$ vertices and $\delta n^2$ edges. Are there subgraphs $H_1,H_2\subseteq G$ such that $H_1$ has $\gg {\delta }^3{n}^2$ edges and every two edges in $H_1$ are contained in a cycle of length at most $6$, and furthermore if two edges share a vertex they are on a cycle of length $4$, and $H_2$ has $\gg {\delta }^2{n}^2$ edges and every two edges in $H_2$ are contained in a cycle of length at most $8$.

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

This is a well-known open Erdős–Duke–Rödl problem (it appears as Erdős Problem #584). As stated, **the full pair of assertions is not known in complete generality**, but there are strong partial results. ([Erdős Problems][1])

llm-hunter · gpt pro 5.2 · unverified

1 LLM attack(s) recorded (gpt pro 5.2); unverified.