Linkage on the infinite grid

For k fixed, a graph G is k-path-pairable, if for any set of k disjoint pairs of vertices, si,ti, 1≤i≤k, there exist pairwise edge-disjoint si,ti-paths in G. Bounds on path-pairability are given here if G is the graph of the infinite integer grid in the Euclidean plane (vertices of G are the points of integer coordinates and two vertices are adjacent if and only if their Manhattan distance is 1). We prove that G is 10-path-pairable and at most 14-path-pairable. Related results and conjectures are summarized also for the integer halfplane, for the positive integer quadrant and for finite grids.