On a Problem of Füredi and Griggs

The Shadow Minimization Problem in the Boolean lattice asks for the minimum cardinality of the shadow of a family of k–sets of [n] among families of the same cardinality. The well–known Kruskal–Katona theorem says that the initial segments \(I_{n,k}(m)\) of length m in the colex order are solutions to the problem. Füredi and Griggs showed that, for some set of cardinalities m, the solution to this problem is unique (up to automorphisms of the Boolean lattice). They gave examples showing that this unicity may fail to hold for other cardinalities and raised the question of characterizing the extremal sets for this problem. We give a structural result for these extremal sets which shows in particular that, for every extremal family S of k–subsets of [n] and every \(t>c\log \log n\), the t iterated lower shadow of S is an initial segment in the colex order. Moreover, for an asymptotically dense set of cardinalities, initial segments in the colex order still are essentially the unique solution to this shadow minimization problem. These results illustrate the robustness of the colex order as a solution of this problem. A key property of the cardinalities for which solutions other than initial segments in the colex order exist is that the coefficients of their k–binomial decompositions decrease very fast, according to a family of numbers which extend a classical sequence of the so–called hypotenusal numbers. We also provide an algorithm linear in n and polynomial in k deciding, given a cardinality m and an integer t, if there is an extremal family S of k–subsets of [n] such that \(\varDelta ^t(S)\) is not an initial segment in the colex order and, if the answer is positive, provides a construction of such a set.