A function is additive if f(x+y)=f(x)+f(y). Intuitively, it might seem that an additive function from R to R must be linear, specifically of the form f(x)=kx. But assuming the axiom of choice, that is wrong, and the proof is rather simple: you just take a Hamel basis of R as a vector space over Q, and then you define your function f to be different in at least two distinct elements of the basis.
But my question is this: if there is no Hamel basis of R, then must f be linear? To put it another way, does ZF + the existence of a nonlinear additive function imply the existence of Hamel basis of R?
I checked the Consequences of the Axiom of Choice Project, a database of choice axioms and their relationships here, and it said that it didn't know.