The following is known as the Efimov problem: is there an infinite compact Hausdorff space which does not contain a nontrivial convergent sequence and does not contain a copy of $\beta N$? Such a space is called an Efimov space. Up till now there are many consistent examples of Efimov spaces but no ZFC example has been constructed and no one proved the consistency of the nonexistence of them. In this talk we give a simple (but not elementary) construction of an Efimov space using Sacks forcing and show some interesting sufficient conditions for a compact space to contain a copy of $\beta N$ (based on the paper P. Koszmider, S. Shelah; Independent families in Boolean algebras with some separation properties. Algebra Universalis 2013)
The problem has a translation to Boolean algebras: "Is there an infinite Boolean algebra without an independent family of cardinality continuum and without an infinite countable Boolean homomorphic image?". The Banach spaces version: "Is there an infinite dimensional nonreflexive Banach space whose quotients do not include spaces isomorphic to $c_0$ nor $l_\infty$" has been established as an undecidable problem (Talagrand, Haydon-Levy-Odell). We plan to discuss these results.