  For a cardinal lambda<lambda_{omega_1} we give a ccc forcing notion P which
forces that for some Borel subset B of the Cantor space
  (1) there a sequence (eta_alpha:alpha<lambda) of distinct elements such that
|(eta_alpha+B) cap (eta_beta+B)|>5 for all alpha,beta<lambda, but
  (2) there is no perfect set of such eta's.
  The construction closely follows the one from Shelah math.LO/9802134
