Arratia (1999) introduced the problem of determining the length of the shortest possible k-superpattern.
[2] He observed that there exists a superpattern of length k2 (given by the lexicographic ordering on the coordinate vectors of points in a square grid) and also observed that, for a superpattern of length n, it must be the case that it has at least as many subsequences as there are patterns.
[2] The upper bound of k2 on superpattern length proven by Arratia is not tight.
[6] Eriksson et al. conjectured that the true length of the shortest k-superpattern is asymptotic to k2/2.
Researchers have also studied the length needed for a sequence generated by a random process to become a superpattern.