Kruskal-Katona function and variants of cross-intersecting antichains

Abstract

We prove some properties of the Kruskal-Katona function and apply to the following variant of cross-intersecting antichains. Let n ≥ 4 be an even integer and A and B be two cross-intersecting antichains on [n] with at most k disjoint pairs, i.e., for all Ai ∈ A , Bj ∈ B, Ai ∩ Bj = ∅ only if i = j ≤ k. We prove a best possible upper bound on |A |+|B| and show that the extremal families contain only <math><mfrac is="true"><mrow is="true"><mi is="true">n</mi></mrow><mrow is="true"><mn is="true">2</mn></mrow></mfrac></math> and <math><mo stretchy="false" is="true">(</mo><mfrac is="true"><mrow is="true"><mi is="true">n</mi></mrow><mrow is="true"><mn is="true">2</mn></mrow></mfrac><mo linebreak="badbreak" linebreakstyle="after" is="true">+</mo><mn is="true">1</mn><mo stretchy="false" is="true">)</mo></math>-sets. The main tools are Sperner operations and Kruskal-Katona’s Theorem.