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Báo cáo toán học: "Low Rank Co-Diagonal Matrices and Ramsey Graphs"

Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: Low Rank Co-Diagonal Matrices and Ramsey Graphs. | Low Rank Co-Diagonal Matrices and Ramsey Graphs Vince Grolmusz Department of Computer Science Eotvos University H-1053 Budapest HUNGARY E-mail grolmusz@cs.elte.hu Submitted March 7 2000. Accepted March 12 2000 Abstract We examinenxn matrices over Zm with 0 s in the diagonal and nonzeros elsewhere. If m is a prime then such matrices have large rank i.e. WAp-1 0 1 . If m is a non-prime-power integer then we show that their rank can be much smaller. For m 6 we construct a matrix of rank exp cựlog n log log n . We also show that explicit constructions of such low rank matrices imply explicit constructions of Ramsey graphs. Keywords composite modulus explicit Ramsey-graph constructions matrices over rings co-diagonal matrices 1 Introduction In this work we examine matrices over a ring R such that the diagonal elements of the matrix are all 0 s but the elements off the diagonal are not zero we shall call these matrices co-diagonal over R . We define the rank of a matrix over a ring and show that low rank codiagonal matrices over Z6 naturally correspond to graphs with small homogenous vertex sets i.e. cliques and anti-cliques . Consequently explicitly constructible low rank co-diagonal matrices over Z6 imply explicit Ramsey graph constructions. Our best construction reproduces the logarithmic order of magnitude of the Ramsey-graph of Frankl and Wilson 5 continuing the sequence of results on new explicit Ramsey graph constructions of Alon 1 and Grolmusz 6 . Our present result analogously to the constructions of 6 and 1 can be generalized to more than one color. 1 THE ELECTRONIC .JOURNAL OF COMBINATORICS 7 2000 R15 2 Our results give a recipe for constructing explicit Ramsey graphs from explicit low rank co-diagonal matrices over Z6 analogously to the way that our results gave a method for constructing explicit Ramsey graphs from certain low degree polynomials over Z6 in 6 . In this sense our results may lead to improved Ramsey graph constructions if lower rank co-diagonal