A flat tax you only start paying above a certain threshold then? For sake of argument everything under 20k is tax free, you pay flat tax on everything above? Problem solved?
I think what msla is saying is that people who are at or below the threshold for taxation would contribute nothing to taxes, which would be unfair in theory because they consume public goods and services like the rest of us.
Remember that Flat Tax is also called Fair Tax. The concept of fairness is central.