problems from the scottish book

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

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.

The second edition of the Mauldin book has information up-to-date as of 2015.