The principle of finite descent

Preliminaries

Select 10 points at random in a unit square :

Permute elements i and j in a list :

Test if segments i (joining red[[i]] and blue[[i]]) and j (joining red[[j]] and blue[[j]]) intersect :

(See Annexe below)

Finite descent (first example)

10 red points and 10 blues points in a unit square :

A pleasant random choice constantly valid in this file :

Initial display :

The 20 points :

One-to-one pairing of the red and blue points (In the order 1-1, 2-2, ..., 10-10) (Total length, L,  of the straight segments displayed) :

Progressive uncrossing of the segments. If the final  uncrossing is incomplete, the cell must be manually reactivated until complete uncrossing is achieved :

Finite descent (second example)

The previous calculation started with bleu=Permutations[blue][[1]] (=blue i.e. Identity permutation). One can do better if starting  with bleu=Permutations[blue][[1066281]] :

New starting point :

Intersection point (X = {xx, yx}) between two straight lines, ab et cd :