A problem of Erdős and Hajnal. The example of $K_{n,n+1}$ shows that this fails if we only have $n^2+n$ edges.

This is true, and was proved by Chen and Ma [ChMa25], who prove the stronger statement that, provided $n\geq 600$, all graphs with $2n+1$ vertices and at least $n^2+n$ edges contain two vertices of the same degree joined by a path of length $3$, except $K_{n,n+1}$.

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