Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Erd˝s-Ko-Rado-Type Theorems for o Colored Sets. | Erdõs-Ko-Rado-Type Theorems for Colored Sets Yu-Shuang Li and Jun Wang y Department of Applied Mathematics Dalian University of Technology Dalian 116024 P. R. China Submitted Jun 7 2006 Accepted Dec 10 2006 Published Jan 3 2007 Mathematics Subject Classihcation 05D05 Abstract An Erdos-Ko-Rado-type theorem was established by Bollobás and Leader for q-signed sets and by Ku and Leader for partial permutations. In this paper we establish an LYM-type inequality for partial permutations and prove Ku and Leader s conjecture on maximal k-uniform intersecting families of partial permutations. Similar results on general colored sets are presented. 1 Introduction Erdos Ko and Rado proved in 1961 10 that a family of pairwise intersecting k-subsets of an n-set cannot have more members than the family of k-subsets all of which contain a given element a say provided k b 2 J. Bollobás in 1973 3 established a stronger result an LYM-type inequality which says that if A is an intersecting antichain of subsets of an n-set then Pk 1 nf 1 where fk denotes the number of sets in A of size k with k n 2. This inequality implies the Erdos-Ko-Rado Theorem. The original LYM inequality says that if A is an antichain of subsets of an n-set then Pn 0 f 1 which yields a simple proof of Sperner s Theorem that A Ik o fk n . This proof is due independently to Lubell Yamamoto and Meschalkin and therefore the inequality is known as the LYM-inequality see 9 for detail . In 1972 Katona presented a rather simple proof of the Erdos-Ko-Rado Theorem. By his technique one can usually establish an LYM-type inequality. By employing Katona s technique in 1997 Bollobas and Leader 4 presented an Erdos-Ko-Rado theorem for q-signed sets where q 2. A q-signed k-set is a pair A f where A c n is a k-set and f Correspondence author E-mail address junwang@ y Supported by the National Natural Science Foundation of China grant number 10471016 . THE ELECTRONIC JOURNAL OF COMBINATORICS 14 2007 R1 1 is a function .