Even unique intersection point can break OSC: an example

This was a long-standing question since 90-ies whether one-point intersection property for a self-similar set implies open set condition. We answer this question negatively. We give an example of a totally disconnected self-similar set K⊂R which does not have open set condition and has minimal overlap of its pieces, that is, all intersections of its pieces Ki ∩ Kj , i 6= j are empty except only one, which is a single point