Another 3-Part Sperner Theorem

In this paper, we prove a higher order Sperner theorem.These theorems are stated after some notation and background results are introduced. For i, j positive integers with i ≤ j, let [i, j] denote the set {i, i + 1, . . . , j}. For k, n positive integers, set ( [n] k ) = {A ⊆ [1, n] : |A| = k}. A system A of subsets of [1, n] is said to be k-set system if A ⊆ ( [n] k ) . Two subsets A, B are incomparable if A 6⊆ B and B 6⊆ A. A set system on an n-set A is said to be a Sperner set system, if any two distinct sets in A are incomparable. Sperner’s Theorem is concerned with the maximal cardinality of Sperner set systems as well as with the structure of such maximal systems.