Extremal Functions for Shortening Sets of Paths

Let P1, . . . , Pk be k vertex disjoint paths in a graph G where the ends of Pi are xi, and yi. Let H be the subgraph induced by the vertex sets of the paths. We find edge bounds E1(n), E2(n) such that, 1. If e(H) ≥ E1(|V (H)|), then there exist disjoint paths P ′ 1, . . . , P ′ k where the ends of P ′ i are xi and yi such that | S