Splitting Numbers of Grids

For a subset S of a finite ordered set P , let S ↑ = {x ∈ P : x ≥ s for some s ∈ S} and S ↓ = {x ∈ P : x ≤ s for some s ∈ S}. For a maximal antichain A of P , let s(A) = max A=U∪D |U ↑ | + |D↓ | |P | , the maximum taken over all partitions U ∪ D of A, and sk(P ) = min A∈A(P ),|A|=k s(A) where we assume P contains at least one maximal antichain of k elements. Finally, for a class C of finite ordered sets, we define sk(C) = inf P∈C sk(P ). Thus sk(C) is the greatest proportion r satisfying: every k-element maximal antichain of a member P of C can be “split” into sets U and D so that U ↑ ∪ D ↓ contains at least r|P | elements. In this paper we determine sk(Gk) for all k ≥ 1, where Gk = {k × n : n ≥ k} is the family of all k by n “grids”. the electronic journal of combinatorics 12 (2005), #R17 1