K-Chains Problem and Why it Matters for Extremal Contexts

Here we discuss a problem of arranging k linear orders on n elements to maximize the number of sets that can be obtained as intersections of their initial intervals. We argue that this problem can shed light on a hard problem of characterizing formal contexts of bounded VC dimension, extremal with respect to the number of their objects and attributes. To tackle this problem we introduce limit objects, which capture their asymptotics, and propose, for all k, a tentative optimal solution. We prove that, under additional hypothesis of symmetry, it is indeed optimal for k = 3.