Rapidly rotating, low-mass members of eclipsing binary systems have measured radii significantly larger than predicted by standard models. It has been proposed that magnetic activity is responsible for radius inflation. By estimating the radii of low-mass stars in three young clusters (NGC 2264, NGC 2547, NGC 2516, with ages of 5, 35 and 140 Myr respectively), we aim to establish whether similar radius inflation is seen in single, magnetically active stars. We use radial velocities from the Gaia-ESO Survey (GES) and published photometry to establish cluster membership and combine GES measurements of vsini with published rotation periods to estimate average radii for groups of fast-rotating cluster members as a function of their luminosity and age. The average radii are compared with the predictions of both standard evolutionary models and variants that include magnetic inhibition of convection and starspots. At a given luminosity, the stellar radii in NGC 2516 and NGC 2547 are larger than predicted by standard evolutionary models at the ages of these clusters. The discrepancy is least pronounced and not significant ~10 percent) in ZAMS stars with radiative cores, but more significant in lower-mass, fully convective pre main-sequence cluster members, reaching 30+/-10 percent. The uncertain age and distance of NGC 2264 preclude a reliable determination of any discrepancy for its members. The median radii we have estimated for low-mass fully convective stars in the older clusters are inconsistent (at the 2-3 sigma level) with non-magnetic evolutionary models and more consistent with models that incorporate the effects of magnetic fields or dark starspots. The available models suggest this requires either surface magnetic fields exceeding 2.5 kG, spots that block about 30 per cent of the photospheric flux, or a more moderate combination of both. [Abridged]
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