  We derive bounds on the number of abelian gauge group factors in
six-dimensional gravitational theories with minimal supersymmetry and in their
F-theoretic realisations. These bounds follow by requiring consistency of
certain BPS strings in the spectrum of the theory, as recently proposed in the
literature. Under certain assumptions this approach constrains the number of
abelian gauge group factors in six-dimensional supergravity theories with at
least one tensor multiplet to be $N \leq 20$ (or $N \leq 22$ in absence of
charged matter). For any geometric F-theory realisation with at least one
tensor multiplet we establish the bound $N \leq 16$ by demanding unitarity of a
heterotic solitonic string which exists even in absence of a perturbative
heterotic dual. This result extends to four-dimensional F-theory vacua on any
blowup of a rational fibration. Our findings lead to universal bounds on the
rank of the Mordell-Weil group of elliptically fibered Calabi-Yau threefolds.
