Recall that a two-point set is a subset of the plane which meets every line in exactly two points. Such a set was first constructed by Mazurkiewicz in 1914.


I wonder if the following question of Erdos is solved:



> 
> **Question.** Is there a two-point set which is Borel?
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> 
> 


See [On sets which meet each line in exactly two points](http://www.math.unt.edu/~mauldin/papers/no100.pdf), where Mauldin says that he “believes” he first heard of the problem from Erdos, who in turn
said that it had been around since he (Erdos) was a “baby.”


**Remarks.** A two-point set can not be $F\_\sigma$ (by a result of Larman). Also it is known that if a two-point set is analytic, then it is Borel.



Can two-point sets be Borel?
A two-point set cannot be $F\_\sigma$, as Mohammad mentions in his question. Also,



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> A two-point set cannot contain a dense $G\_\delta$ subset of an arc.
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This was proved by Gareth Davies in his thesis (Oxford, 2011), but I do not think he ever published this result. To my knowledge, no better results are known in the Borel-sets-shouldn't-work direction.


In the other direction, the best known results belong to Arnie Miller. He showed that



> 
> If $V=L$ then there is a co-analytic two-point set.
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> 
> 


See [this](http://www.math.wisc.edu/~miller/res/infcomb.pdf) paper of his from 1989 where he writes about this question and other such things, like nicely-definable Hamel bases and MAD families. Also see [this](https://arxiv.org/pdf/1209.4267.pdf) related paper of Zoltan Vidnyansky's, which appeared more recently and extended/streamlined some of Miller's work.


In an unpublished manuscript that he made available [here](http://www.math.wisc.edu/~miller/res/two-pt.pdf) on his website, Arnie Miller also showed that



> 
> It is consistent for the Axiom of Choice to fail badly, but still to have two-point sets (e.g., two-point sets can exist even when there is no well-ordering of the reals).
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> 
> 


Ben Chad did quite a bit of work about 5-10 years ago in trying to eliminate the Axiom of Choice from constructions of two-point sets as much as possible. Some of this stuff made it into [this paper](https://people.maths.ox.ac.uk/chad/Papers/setTheoreticConstructionsOfTwoPointSets.pdf), joint with Robin Knight and Rolf Suabedissen.


