tailieunhanh - Báo cáo toán học: "Note on generating all subsets of a finite set with disjoint union"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Note on generating all subsets of a finite set with disjoint unions. | Note on generating all subsets of a finite set with disjoint unions David Ellis e-mail dce27@ Submitted Dec 2 2008 Accepted May 12 2009 Published May 20 2009 Mathematics Subject Classification 05D05 Abstract We call a family G c P n a k-generator of P n if every x c n can be expressed as a union of at most k disjoint sets in G. Frein Leveque and Sebo 1 conjectured that for any n k such a family must be at least as large as the k-generator obtained by taking a partition of n into classes of sizes as equal as possible and taking the union of the power-sets of the classes. We generalize a theorem of Alon and Frankl 2 in order to show that for fixed k any k-generator of P n must have size at least k2n k 1 o 1 thereby verifying the conjecture asymptotically for multiples of k. 1 Introduction We call a family G c P n a k-generator of P n if every x c n can be expressed as a union of at most k disjoint sets in G. Frein Leveque and Sebo 1 conjectured that for any n k such a family must be at least as large as the k-generator k Fn k u PVi 0 1 i 1 where Vi is a partition of n into k classes of sizes as equal as possible. For k 2 removing the disjointness condition yields the stronger conjecture of Erdos - namely if G c P n is a family such that any subset of n is a union not necessarily disjoint of at most two sets in G then G is at least as large as F . PV1 u PV2 0 2 where V1 V2 is a partition of n into two classes of sizes I_n 2J and n 2 . We refer the reader to for example Ftiredi and Katona 5 for some results around the Erdos conjecture. In fact Frein Leveque and Sebo 1 made the analagous conjecture for all k. We call a THE ELECTRONIC JOURNAL OF COMBINATORICS 16 2009 N16 1 family G c P n a k-base of P n if every x c n can be expressed as a union of at most k sets in G they conjectured that for any k n any k-base of P n is at least as large as Fn k. In this paper we show that for k fixed a k-generator must have size at least k2n k 1 o 1 when n is a multiple of k