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For positive integers d and m, let $P_{d,m}\left(G\right)$ denote the property that between each pair of vertices of the graph G, there are m internally vertex disjoint paths of length at most d. For a positive integer t a graph G satisfies the minimum generalized degree condition δₜ(G) ≥ s if the cardinality of the union of the neighborhoods of each set of t vertices of G is at least s. Generalized degree conditions that ensure that $P_{d,m}\left(G\right)$ is satisfied have been investigated. In particular, it has been shown, for fixed positive...