Extremal Configurations on a Discrete Torus and a Generalization of the Generalized Macaulay Theorem

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Let $V = \sum\nolimits_{i = 1}^n {Z / (k_i + 1)},0\leqq k_1 \leqq k_2 \leqq \cdots \leqq k_n \leqq \infty $, be the set of discrete lattice points in a torus. Let $e_i $ be the unit vector along the ith axis, $i = 1,2, \cdots ,n,$ and let ${\operatorname{Bdry}}(A) = \{ x \in V| {x \notin A,x - e_i \in A\,\,{\text{for some}}\,i} \} $. In this paper we obtain extremal configurations that minimize the number of boundary points among sets of given size in V. We show further that the result obtained implies the generalized Macaulay theorem of Clements and Lindström and the well-known Kruskal–Katona theorem.