Using new and archival observations made with the Swift satellite and other facilities, we examine 147 X-ray sources selected from the ROSAT All-Sky-Survey Bright Source Catalog (RASS/BSC) to produce a new limit on the number of isolated neutron stars (INSs) in the RASS/BSC, the most constraining such limit to-date. Independent of X-ray spectrum and variability, the number of INSs is <=48 (90% confidence). Restricting attention to soft (having an effective temperature of < 200 eV), non-variable X-ray sources -- as in a previous study -- yields an all-sky limit of <=31 INSs. In the course of our analysis, we identify five new high-quality INS candidates for targeted follow-up observations. A future all-sky X-ray survey with eROSITA, or another mission with similar capabilities, can be expected to increase the detected population of X-ray-discovered INSs from the 8 to 50 in the BSC, to (for a disk population) 240 to 1500, which will enable a more detailed study of neutron star population models.
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