Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: Frankl-F¨ redi Type Inequalities for Polynomial Semi-lattices u Jin. | Frankl-Fiiredi Type Inequalities for Polynomial Semi-lattices Jin Qian and Dijen K. Ray-Chaudhuri1 Department of Mathematics The Ohio State University Submitted April 2 1997 Accepted October 20 1997 Abstract Let X be an n-set and L a set of nonnegative integers. F a set of subsets of X is said to be an L -intersection family if and only if for all E F 2 F E 0 FI 2 L. A special case of a conjecture of Frankl and Ftiredi 4 states that if L 1 2 . kg k a positive integer then F pk 0 n x . Here F denotes the number of elements in F. Recently Ramanan proved this conjecture in 6 We extend his method to polynomial semi-lattices and we also study some special L-intersection families on polynomial semi-lattices. Finally we prove two modular versions of Ray-Chaudhuri-Wilson inequality for polynomial semi-lattices. 1. Introduction Throughout the paper we assume k n 2 N In 1 2 . ng c N where N denotes the set of positive integers. In this part we briefly review the concept of polynomial semi-lattice introduced by Ray-Chaudhuri and Zhu in 8 The definition of polynomial semi-lattice given here is equivalent to but simpler than that in 8 . For the convenience of the reader we also include various examples of polynomial semi-lattices. Let X be a finite nonempty partially ordered set having the property that X is a semi-lattice . for every x y 2 X there is a unique greatest lower bound of x and y denoted by x A y. If x y and x y we write x y. We 1e-mail addresses qian@ dijen@ THE ELECTRONIC JOURNAL OF COMBINATORICS 4 1997 R28 2 also assume that X has a height function l x where l x 1 is the number of terms in a maximal chain from the least element 0 to the element x including the end elements in the count. Let n be the maximum of l x for all the x in X. Dehne Xi x 2 XI l x i 0 i n and XQ 0 . Then X U qX is a partition and the subsets Xi s are called hbres. The integer n is said to be the height of X . X is called a polynomial semi-lattice if
