A special case of a theorem of Brian Scott (from [*On the existence of totally inhomogeneous spaces*](https://www.ams.org/journals/proc/1975-051-02/S0002-9939-1975-0375262-5/S0002-9939-1975-0375262-5.pdf)) is that there is a size-continuum set $S\subset\mathbb{R}$ such that if $x,y\in S$ are distinct then $S\setminus\{x\}\not\cong S\setminus\{y\}$. The proof relies heavily on the axiom of choice, and at a glance any such $S$ must be quite wild. However, I don't immediately see how to show that such an $S$ must fail any of the standard regularity properties (e.g. measurability).



> 
> **Question**: Is the existence of such an $S$ consistent with $\mathsf{ZF+DC+AD}$?
> 
> 
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"Obviously" the answer is negative ...



Can totally inhomogeneous sets of reals coexist with determinacy?
In *Rigid Borel sets and better quasiorder theory* (Logic and combinatorics, Proc. AMS-IMS-SIAM Conf., Arcata/Calif. 1985, Contemp. Math. 65, 199-222 (1987), [zbMath review here](https://zbmath.org/0646.03045)) Fons van Engelen, Arnold Miller, and John Steel showed that the only rigid Borel sets are the singletons; the proof yields the same for all sets of reals if one assumes $\mathsf{AD}$.
So the answer to the present question is negative.


