A complement of the Erdős-Hajnal problem on paths with equal-degree endpoints

Answering a question of Erdős and Hajnal, Chen and Ma proved that for all $n\ge 600$ every graph with $2n+1$ vertices and at least $n^2+n+1$ edges contains two vertices of equal degree connected by a path of length three. The complete bipartite graph $K_{n,n+1}$ shows that this edge bound is sharp. In this paper, we develop a novel approach to handle graphs with large equal degrees, which enables us to establish the result for all $n\ge 2$, thereby fully resolving the problem posed by Erdős and Hajnal.