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We provide a short proof that if κ is a regular cardinal with κ < c, then
for any ordinal α < min{
, κ}. In particular,
for any ordinal α < . This generalizes an unpublished result of E. Szemerédi that Martin's axiom implies that
for any cardinal κ with κ < c.
Working in the theory ”ZF + There is a nontrivial elementary embedding j : V → V“, we show that a final segment of cardinals satisfies certain square bracket finite and infinite exponent partition relations. As a corollary to this, we show that this final segment is composed of Jonsson cardinals. We then show how to force and bring this situation down to small alephs. A prototypical result is the construction of a model for ZF in which every cardinal μ ≥ ℵ2 satisfies the square bracket infinite exponent partition relation 
. We conclude with a discussion of some consistency questions concerning different versions of the axiom asserting the existence of a nontrivial elementary embedding j: V → V. By virtue of Kunen's celebrated inconsistency result, we use only a restricted amount of the Axiom of Choice.
