  Local sentences were introduced by J.-P. Ressayre who proved certain
remarkable stretching theorems establishing the equivalence between the
existence of finite models for these sentences and the existence of some
infinite well ordered models. Two of these stretching theorems were only proved
under certain large cardinal axioms but the question of their exact
(consistency) strength was left open in [O. Finkel and J.-P. Ressayre,
Stretchings, Journal of Symbolic Logic, Volume 61 (2), 1996, p. 563-585 ].
Here, we solve this problem, using a combinatorial result of J. H. Schmerl. In
fact, we show that the stretching principles are equivalent to the existence of
n-Mahlo cardinals for appropriate integers n. This is done by proving first
that for each integer n, there is a local sentence phi_n which has well ordered
models of order type alpha, for every infinite ordinal alpha > omega which is
not an n-Mahlo cardinal.
