We present six data examples where we infer erosion of the dayside magnetosphere during intervals of very tenuous solar wind (density < 1 cm−3). The interplanetary observations were made by the Wind spacecraft when the average solar wind dynamic pressure Pdyn and the interplanetary magnetic field (IMF) Bz were in the ranges (0.07, 0.62) nPa and (−7.6, −0.9) nT, respectively. The inner magnetospheric signature of erosion we focus on is a decrease in the strength of the geostationary magnetic field, as monitored by NOAA's GOES spacecraft. We obtain this decrease as a function of IMF Bz by comparing each event with a reference day, May 11, 1999. During the reference day the lowest Pdyn of the set was attained (0.07 nPa), IMF Bz > 0, and the geomagnetic field at geostationary orbit was dipolar. The central point we make is that although compared to the reference day the Pdyn in each event is higher, the strength of the geostationary field is weaker. We interpret this as evidence that the field compression due to Pdyn has been overcome by the field depression due to erosion. Correcting empirically for the compression of the geostationary field due to solar wind dynamic pressure, we find that for the tenuous solar winds we consider the decrease of the geostationary field, ΔBGS, is related to IMF Bz as ΔBGS (nT)= −2.8 + 2.3 Bz (nT).