The exoplanet population in close orbits (semi-major axis a<0.4 AU) around solar-type stars (FGK dwarfs) is dominated by small planets (SPs) with mass Mp < 20 Mearth, i.e. super-Earths and sub-Neptunes. These planets are, however, missing in our Solar System, and the reason for that is unknown. By studying the impact of cold Jupiters (CJs) with a=1-10 AU and Mp=0.5-20 MJup on the formation and/or migration of SPs, several theoretical works have predicted either an anti-correlation or a weak or strong correlation between CJs and SPs, thus reaching somehow contradictory results. Here we report on the occurrence rate of CJs in the largest sample of SP systems ever considered, using high-precision radial velocities. We find an integrated occurrence rate of CJs in small-planet systems of f(CJ|SE)~11%, which is considerably lower than previous estimates reported by other groups, and a possible excess of CJs only at super-solar mass and metallicity. By investigating the SP-CJ relation with CJ properties, SP multiplicity, and SP composition, we provide important clues on both the formation of short-period SPs and their absence in our Solar System.