turkmath.org

Make the set of subsets of a given set into a metric space by defining the distance between two subsets as the cardinality of their symmetric difference. Say elements in a metric space form a system without flat triangles if triangular inequality between any three of them is always strict. We will be interested in giving asymptotic bounds, for n going to infinity, on the maximal cardinality of a system without flat triangles in the set of subsets of an n element set. A lower bound, purely existential, can be given using the probabilistic method (à la Erdös). We will improve this bound, constructively, with a completely different method using ideas from the theory of error correcting codes and new results in algebraic geometry.