We say that a σ-ideal I on a Polish space X has the "1-1 or constant" property if every Borel function defined on a Borel I-positive subset of X can be restricted to a Borel I-positive set, on which it is 1-1 or constant. In other words, this means that the forcing P_I adds a minimal real degree. The classical well-known examples of σ-ideals with the 1-1 or constant property are the σ-ideal of countable sets (Sacks forcing) and the σ-ideal of σ-compact subsets of the Baire space (Miller forcing). On the other hand, an example for which this property drastically does not hold is the σ-ideal of meager sets (or any I for which P_I adds a Cohen real). During the talk I will prove the following dichotomy: if I is a σ-ideal generated by closed sets, then

(i) either I has the 1-1 or constant property,<br/> (ii) or else P_I adds a Cohen real.

This is joint work with Jindra Zapletal.