Some structural properties of low rank matrices related

We consider the problem of the presence of short cycles in the graphs of nonzero elements of matrices which have sublinear rank and nonzero entries on the main diagonal, and analyze the connection of these properties to the rigidity of matrices. In particular, we exhibit a family of matrices which shows that sublinear rank does not imply the existence of triangles. This family can also be used to give a constructive bound of the order of k 3=2 on the Ramsey number R(3; k), which matches the best known bound. On the other hand we prove that sublinear rank implies the existence of 4-cycles. We prove some partial results towards establishing lower bounds on matrix rigidity and consequently on the size of logarithmic depth circuits for computing certain explicit linear transformations.