As is known, Tarski posed his questions about first-order theories of non-abelian free groups around 1945. However, the questions were not published in his papers or books.
What is the original published source of reliable information about posing Tarski's problems on free groups? When exactly had Tarski posed these questions at first time? Did it happen at some seminar or conference?
Tarski presented these problems at the Bicentennial Conference on Problems of Mathematios at Princeton, on december 1946. There's a relatively recent exposition of the material by Hourya Sinaceur:
Address at the Princeton University Bicentennial Conference on Problems of Mathematics (December 17-19, 1946) The Bulletin of Symbolic Logic Vol. 6, No. 1 (Mar., 2000), pp. 1-44
No notes of the conference were published at the time. If he wrote anything else about this before 1946, or shortly after, it most likely remained unpublished (perhaps you can find something in his collected works).
Why did I guess that Tarski's problems about first-order theories of non-abelian free groups were not published in his papers or books, and he posed the problems around 1945? Here are the reasons of that:
$\bullet$ In the papers of Kharlampovich-Myasnikov [J. Algebra 302 (2006), no. 2, 451-552] and Sela [Geom. Funct. Anal. 16 (2006), no. 3, 707-730] with solutions of Tarski's problems there is no references to his publications;
$\bullet$ In the Kharlampovich-Myasnikov paper it is written that the Tarski's conjectures were formulated around 1945;
$\bullet$ Lyndon in his paper [Problems in combinatorial group theory, in: Combinatorial group theory and topology, Ann. Math. Stud., Princeton Univ. Press, 1987] formulated the problem on elementary equivalence of all non-abelian free groups and called it a folklore problem of Alfred Tarski.
I checked Tarski's papers and books and now I know that his problems on the elementary theory of free groups had been formulated in some of his publications; so these problems were not `folklore'. Also, it seems the problems had been posed not around 1945 but later.
The first Tarski's publication where the decision problem for the theory of non-abelian free groups was mentioned is the Tarski's book [Undecidable theories, North-Holland, 1953]. On p. 77 he informed that he stated his result of undecidability of the elementary theory of groups at a conference in Princeton in December, 1946. On p. 85, he wrote: 'For many extensions of the elementary theory of groups, e.g. for elementary theories of finite groups and non-Abelian free groups, the decision problem remains open'. Note that it was not a conjecture on decidability of the elementary theory of non-abelian free groups; he just said that it was not known whether the theory is decidable or not. Clearly, the question had been raised between 1945 and 1953, and I don't know when exactly.
In that Tarski's book the problem of elementary equivalence of all non-abelian free groups was not mentioned. First time it was published in Vaught's abstract [Bull. AMS, 61 (1955), N 2, 173-174], where he formulated his theorem implying that standard embeddings of free groups of infinite rank are elementary and wrote: 'These investigations arose from a still unresolved conjecture of Tarski's that any two free groups with at least two generators are arithmetically equivalent'. That theorem was a result from Chapter 3 of Vaught's PhD thesis (Berkeley, 1954), the supervisor of which was Tarski. Later the Vaught's theorem and the Tarski's conjecture were published in Tarski-Vaught's paper [Compositio Math. 13 (1957), 81-102]; see pages 82 and 98. Also, in the introduction of the paper it is written that the decision problem for the elementary theory of free non-abelian groups is a closely related open problem.
