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.

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