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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?

See On sets which meet each line in exactly two points, 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.

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    $\begingroup$ I think it is still open as per a search through google scholar. For one such example, see footnote 3 in Chad & Good's (2011) Homeomorphisms of Two-Point Sets in the Proceedings of the AMS (pdf). $\endgroup$
    – Benjamin Dickman
    Jun 19, 2017 at 7:35
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    $\begingroup$ For people curious about constructions of two-point set, I'll add a link to an older question: Subset of the plane that intersects every line exactly twice. $\endgroup$
    – Martin Sleziak
    Jun 19, 2017 at 9:08
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    $\begingroup$ I once tried to solve this problem, starting with the first unknown case: to show that such a set can't be G_delta. I'd be interested to know even whether that problem is solved. $\endgroup$
    – gowers
    Jun 19, 2017 at 10:47
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    $\begingroup$ See also this related question, mathoverflow.net/q/93601/1946, which asks the corresponding question for other geometric constructions that can be undertaken using the axiom of choice, but where it is not clear whether one can find explicit or Borel instances. $\endgroup$
    – Joel David Hamkins
    Jun 19, 2017 at 13:25
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    $\begingroup$ This is problem AM in Fremlin's problem list (essex.ac.uk/maths/people/fremlin/problems.pdf) as of February 2016. He seems to credit the question to Sierpinski. And he's offering 34 British pounds for a solution, in case you weren't already interested enough! $\endgroup$
    – Paul McKenney
    Jun 19, 2017 at 19:15

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A two-point set cannot be $F_\sigma$, as Mohammad mentions in his question. Also,

A two-point set cannot contain a dense $G_\delta$ subset of an arc.

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.

See this 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 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 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).

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, joint with Robin Knight and Rolf Suabedissen.

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