Abstract: I will discuss new constraints on 6d (1,0) supergravity theories derived from consistency conditions, including the positivity of the metric and string tensions, within the Kähler moduli spaces of their 5d reductions. The emergence of local 5d CFTs, combined with the BPS spectra of 2d strings, imposes further constraints. I will demonstrate how these conditions can be used to restrict the gauge groups and matter charges in a class of 6d supergravity theories with rank 1 gauge algebras.