We deduce an expression for the critical polarization P _ { C } below which the FFLO-state emerges in one-dimensional lattices with spin-imbalanced populations .
We provide and explore the phase diagram of unconfined chains as a function of polarization , interaction and particle density .
For harmonically confined systems we supply a quantitative mapping which allows to apply our phase diagram also for confined chains .
We find analytically , and confirm numerically , that the upper bound for the critical polarization is universal : P _ { C } ^ { max } = 1 / 3 for any density , interaction and confinement strength .