Patterns that can be avoided by large sets in R^n

Some "patterns" universally appear in any large measurable set in R^n, and the others can be avoided by such sets. (By "large" we mean a set of positive measure, or Borel set of sufficiently large Hausdorff dimension, etc.) Trivial examples of such results are that any set of positive measure realizes or sufficiently small distances by pairs of points in the set, or that any such set contains many copies geometrically similar to any prescribed finite set. On the other hand, even in R, a set of positive measure can avoid to contain a homothetic image of the sequence {1/n: n positive integer}.

For a particular "pattern" (and the notion of "large") to recognize if it is avoidable, or if it universally appears in each large measurable set, is typically very difficult question. Some new results and many open questions are discussed.