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Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: On the number of orthogonal systems in vector spaces over finite fields. | On the number of orthogonal systems in vector spaces over finite fields Le Anh Vinh Mathematics Department Harvard University Cambridge MA 02138 US vinh@math.harvard.edu Submitted Jul 15 2008 Accepted Aug 13 2008 Published Aug 25 2008 Mathematics Subject Classification 05C50 05C35 Abstract losevich and Senger 2008 showed that if a subset of the d-dimensional vector space over a finite field is large enough then it contains many k-tuples of mutually orthogonal vectors. In this note we provide a graph theoretic proof of this result. 1 Introduction A classical set of problems in combinatorial geometry deals with the question of whether a sufficiently large subset of Rd zd or Fd contains a given geometric configuration. In a recent paper 3 losevich and Senger showed that a sufficiently large subset oCFd the d-dimensional vector space over the finite field with q elements contains many k-tuple of mutually orthogonal vectors. Using geometric and character sum machinery they proved the following result see 3 for the motivation of this result . Theorem 1.1 3 Let E c Fd such that EI Cqd M 1 1.1 with a sufficiently large constant C 0 where 0 2 d. Let Afc be the number of k-tuples of k mutually orthogonal vectors in E. Then k 1 o 1 - . 1.2 k In this note we provide a different proof to this result using graph theoretic methods. The main result of this note is the following. THE ELECTRONIC JOURNAL OF COMBINATORICS 15 2008 N32 1 Theorem 1.2 Let E c Fd such that IE 1 qi 1-1 1.3 where d 2 k 1 . Then the number of k-tuples of k mutually orthogonal vectors in E is 1 o 1 11 q- . 1.4 k Note that Theorem 1.1 only works in the range d 2 as larger tuples of mutually orthogonal vectors are out of range of the methods uses while Theorem 1.2 works in a wider range d 2 k 1 . Moreover Theorem 1.2 is stronger than Theorem 1.1 in the same range. 1.1 Sharpness of results It is also interesting to note that the exponent d 1 cannot be improved in the case k 2. In 3 losevich and Senger constructed
