problems from the scottish book Which of the problems from the Scottish Book (pdf of English version) by Stefan Banach are still open? I know that one of the problems was solved by Per Enflo for which he got a live goose from Stanislaw Mazur.
 A: The book version edited by Daniel Mauldin (from 1982) has commentaries on the problems as of that date.
A: Luis Montejano solved problem 68 in 1990 (he also solved the limiting 0 density case of problem 19, but I think this is already mentioned in Mauldin's book). The paper is called About a problem of Ulam concerning flat sections of manifolds and appeared in Commentarii Mathematici Helvetici.
A: The second edition of the Mauldin book has information up-to-date as of 2015.
A: Second problem in The Scottish Book (Edit: is open? GRP). Let $X$ be a compact metric space. If there exists a finitely additive Borel measure $\mu$ such that $\mu(X)=1$ and if $A,B\subset X$ are congruent then $\mu(A)=\mu(B)$.
Remark. We say that that the sets $A,B \subset X$ are $congruent$ if there exists a distance preserving bijection from $A$ to $B$, not necessarily defined on the whole space $X$.
2nd edit. I think the best reference frame, related to the above mentioned problem, is a book by Stan Wagon "The Banach-Tarski Paradox". On page 31 of this book there is described recent progress (up to 1985) toward a solution of the problem.
