I gave a plenary talk at the 2017 ASL North American Meeting in Boise, March 2017.
Talk Title: The current state of the Souslin problem.
Abstract: Recall that the real line is that unique separable, dense linear ordering with no endpoints in which every bounded set has a least upper bound.
A problem posed by Mikhail Souslin in 1920 asks whetherthe term separable in the above characterization may be weakened to ccc. (A linear order is said to be separable if it has a countable dense subset. It is said to be ccc if every pairwise-disjoint family of open intervals is countable.)
Amazingly enough, the resolution of this single problem lead to key discoveries in Set Theory: the notions of Aronszajn, Souslin and Kurepa trees, forcing axioms and the method of iterated forcing, Jensen’s diamond and square principles, and the theory of iteration without adding reals.
Souslin problem is equivalent to the existence of a partial order of size $\aleph_1$.
A generalization of this problem to the level of $\aleph_2$ has been identified in the early 1970’s, and is open ever since. In the last couple of years, a considerable progress has been made on the generalized Souslin problem and its relatives. In this talk, I shall describe the current state of this research.