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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.

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One of the problems is the maximum determinant problem. In addition to the references in Mauldin's book, Will Orrick's website maxdet.indiana.edu and recent work posted on the arXiv by Brent and Osborn should help get you up to date. Gerhard "Ask Me About Determinant Spectrum" Paseman, 2013.02.17 –  Gerhard Paseman Feb 18 '13 at 4:27
    
@GerhardPaseman the correct link is indiana.edu/~maxdet –  Federico Poloni Feb 18 '13 at 23:37
    
Thank you, Federico. Gerhard "It Was Something Like That" Paseman, 2013.02.18 –  Gerhard Paseman Feb 19 '13 at 6:24
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3 Answers

up vote 4 down vote accepted

The book version edited by Daniel Mauldin (from 1982) has commentaries on the problems as of that date.

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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.

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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.

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It would be nice to add clarity. I am guessing the problem is open, but a link to a recent mention in the literature would come much closer to answering the spirit of the poster's question. Gerhard "Or Is The Problem Closed?" Paseman, 2013.02.18 –  Gerhard Paseman Feb 18 '13 at 22:40
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