For every n at least 2, every (2n+1)-vertex graph with at least n^2+n+1 edges contains two equal-degree vertices joined by a path of length three, with K_{n,n+1} as the unique extremal graph.
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A complement of the Erd\H{o}s-Hajnal problem on paths with equal-degree endpoints
For every n at least 2, every (2n+1)-vertex graph with at least n^2+n+1 edges contains two equal-degree vertices joined by a path of length three, with K_{n,n+1} as the unique extremal graph.