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

We address a problem posed by Erdős and Hajnal in 1991, proving that for all n ≥ 600 , every ( 2 n + 1 ) -vertex graph with 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 demonstrates that this edge bound is sharp. We further establish an analogous result for graphs with even order and investigate several related extremal problems.