Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: Reconstructing subsets of reals . Radcliffe 1 Department of Mathematics and Statistics. | Reconstructing subsets of reals . Radcliffe 1 Department of Mathematics and Statistics University of Nebraska-Lincoln Lincoln NE 68588-0323 jradclif@ . Scott Department of Mathematics University College Gower Street London WC1E 6BT Submitted November 24 1998 Accepted March 15 1999 1Partially supported by NSF Grant DMS-9401351 Abstract We consider the problem of reconstructing a set of real numbers up to translation from the multiset of its subsets of fixed size given up to translation. This is impossible in general for instance almost all subsets of z contain infinitely many translates of every finite subset of z. We therefore restrict our attention to subsets of R which are locally finite those which contain only finitely many translates of any given finite set of size at least 2. We prove that every locally finite subset of R is reconstructible from the multiset of its 3-subsets given up to translation. Primary 05E99 Secondary 05C60 1 Introduction. Reconstructing combinatorial objects from information about their subobjects is a long-standing problem. The Reconstruction Conjecture and the Edge Reconstruction Conjecture both deal with the problem of reconstructing a graph from a multiset of subgraphs in one case the collection of all induced subgraphs with one fewer vertex in the other the collection of all subgraphs with one fewer edge see Bondy 2 and Bondy and Hemminger 3 f _ The very general problem is that of reconstructing a combinatorial object up to isomorphism from the collection of isomorphism classes of its subobjects. Isomorphism plays a crucial role. Thus it seems that the natural ingredients for a reconstruction problem are a group action to provide a notion of isomorphism and an idea of what constitutes a subobject. Reconstruction problems have been considered from this perspective by for instance Alon Caro Krasikov and Roditty 1 Radcliffe and Scott 11 10 Cameron 4 5 and Mnukhin 7 8 9 . In this paper we consider .
