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Full Dimensional Sets Without Given Patterns

We construct a d Hausdorff dimensional compact set in Rd that does not contain the vertices of any parallelogram. We also prove that for any given triangle (3 given points in the plane) there exists a compact set in R2 of Hausdorff dimension 2 that does not contain any similar copy of the triangle. On the other hand, we show that the set of the 3-point patterns of a 1-dimensional compact set of R is dense.