Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: Splitting Numbers of Grids Dwight Duffus. | Splitting Numbers of Grids Dwight Duffus Mathematics and Computer Science Department Emory University Atlanta GA 30322 UsA dwight@ Bill Sands Mathematics and Statistics Department The University of Calgary Calgary AB T2N 1N4 cAnAdA sands@ Submitted Nov 10 2003 Accepted Apr 4 2005 Published Apr 13 2005 MR Subject Classifications 2000 06A07 06D99 Abstract For a subset S of a finite ordered set P let S t fx 2 P x s for some s 2 S and S ị fx 2 P x s for some s 2 S . For a maximal antichain A of P let _ Utl Dị I s A max -------- I------ v 7 A U D P I the maximum taken over all partitions U u D of A and sk P min s A A2A P A k 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 P2C 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 t u D ị contains at least r PI elements. In this paper we determine sk Gk for all k 1 where Gk fk X n n k is the family of all k by n grids . THE ELECTRONIC JOURNAL OF COMBINATORICS 12 2005 R17 1 1 Introduction Given a maximal antichain A of an ordered set P say that A splits if there is a partition A U u D such that P U t u D ị where U t x 2 P x u for some u 2 U and D ị x 2 P x d for some d 2 D . Say that P has the splitting property if every maximal antichain of P splits. Ahlswede Erdos and Graham introduced these notions in 1 and proved that every finite Boolean lattice has the splitting property. In 2 we used the splitting property to study maximal antichains in distributive lattices. More recently in 3 we characterized the set of distributive lattices with the splitting property and also introduced the idea of a splitting number for any finite ordered set and any class of finite ordered sets. We restate the required definitions. For a maximal antichain A of a finite ordered set P let U tl D ị I s A max -------7 ------
