e.g. The story you have just told me ...
'the story' is an antecedent; it's an object to 'have told'.
Nothing I have heard... - the same.
An instrument we use ... - the same.
#2 'people offering you some money' is a different construction.
'Offering you some money' is not a clause, but a participial phrase, also functioning as an attribute. You can transform it into a clause 'who are offering you ...', but people like to express their ideas in a compressed way. They like to be economical.
Student or Learner