A complement of the Erd\H{o}s-Hajnal problem on paths with equal-degree endpoints

Answering a question of Erd\H{o}s and Hajnal, Chen and Ma proved that for all $n\geq600$ 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 $n\ge2$, thereby fully resolving the problem posed by Erd\H{o}s and Hajnal.