TAILIEUCHUNG - Báo cáo toán học: "On a Conjecture of Frankl and F¨redi u"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: On a Conjecture of Frankl and F¨redi u. | On a Conjecture of Frankl and Fiiredi Ameera Chowdhury Submitted Nov 8 2010 Accepted Feb 27 2011 Published Mar 11 2011 Mathematics Subject Classification 05D05 Abstract Frankl and Furedi conjectured that if F c 2X is a non-trivial A-intersecting family of size m then the number of pairs x y E X that are contained in some F E F is at least m P. Frankl and Z. Furedi. A Sharpening of Fisher s Inequality. Discrete Math. 90 1 103-107 1991 . We verify this conjecture in some special cases focusing especially on the case where F is additionally required to be k-uniform and A is small. 1 Introduction Let X be an n-element set and let 2X denote the family of all subsets of X. For 0 k n let X denote the family of all subsets of X of cardinality k. For a family F c 2X we define the i-shadow of F denoted BZF to consist of those i-subsets of X contained in at least one member of F BiF E E X E c F E F ị . i A family F c 2X is called A-intersecting if we have F1 n F2 A for any distinct F1 F2 E F. The well-known Fisher s Inequality states that if F is a A-intersecting family of size m then Ỡ1F m. Inspired by Fisher s Inequality Frankl and Furedi 7 conjectured a similar inequality for Ỡ2F . Conjecture easily implies Fisher s Inequality since f1 2F Ợm proves Ỡ1F m. Conjecture Frankl-Furedi 1991 Let F c 2X be a A-intersecting family of size m. If there does not exist x E X such that x E F for all F E F then d2F m . Department of Mathematics University of California San Diego La Jolla CA 92093 USA. E-mail anchowdh@ THE ELECTRONIC JOURNAL OF COMBINATORICS 18 2011 P56 1 Frankl and Fiiredi 7 verified Conjecture when A 1. While this paper appears to be the first to consider Conjecture since 7 several special cases of Conjecture had already been proved before 7 was published. For example Ryser 13 Woodall 16 and Babai 1 showed Conjecture is true when m n. Majindar 11 proved Conjecture for regular A-intersecting families. Notation We say F is .

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