For every fixed odd integer k at least 3 and all sufficiently large n, every graph on 2n+1 vertices with n squared plus n plus 1 edges contains two equal-degree vertices joined by a path of length k.
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A generalization of Erd\H{o}s-Hajnal problem on paths with equal-degree endpoints
For every fixed odd integer k at least 3 and all sufficiently large n, every graph on 2n+1 vertices with n squared plus n plus 1 edges contains two equal-degree vertices joined by a path of length k.