Induced matchings in point-tube incidence graphs

Consider the incidence graph between the points of the unit square and the set of δ x 1 tubes for some δ > 0. What is the size n = n(δ) of the largest induced matching in this graph? We use ideas from projection theory to study this problem and show a non-trivial upper bound on n(δ). As a consequence, we derive a new upper bound on the Heilbronn's triangle problem: any set of n points in the unit square contains a triangle of area at most n^{-7/6}. Joint work with Alex Cohen and Cosmin Pohoata.