Cardinal numbers p and t are equal

You need to know: Notation ${\mathbb N}$ for the set of natural numbers. Cardinality of a set: sets A and B have the same cardinality (we write $|A|=|B|$ ) if there exists a bijection from A to B. We write $|A|\leq |B|$ if there exists a bijection from A to a subset of B. It is known that $|A|\leq |B|$ or $|B|\leq |A|$ for every $A,B$ .

Background: Let $A\subseteq^*B$ mean that set $\{x: x\in A, x\not\in B\}$ is finite. Let ${\mathbb N}^{\mathbb N}$ be the family of all infinite sequences of natural numbers, and let $D\subset {\mathbb N}^{\mathbb N}$ . We say that D has a pseudo-intersection if there is an infinite $A \subseteq {\mathbb N}$ such that $A\subseteq^*B$ for all $B \in D$ . We say that D has the strong finite intersection property (s.f.i.p. in short) if every nonempty finite subfamily of D has infinite intersection. D is called well ordered by $\subseteq^*$ if $A\subseteq^*B$ or $B\subseteq^*A$ for all $A,B \in D$ . D is called a tower if it is well ordered by $\subseteq^*$ and has no pseudo-intersection. Let $\textbf{p}$ be the smallest cardinality of $D\subset {\mathbb N}^{\mathbb N}$ which has the s.f.i.p. but has no pseudo-intersection. Let $\textbf{t}$ be the smallest cardinality of $D\subset {\mathbb N}^{\mathbb N}$ which is a tower.

The Theorem: On 27th August 2012, Maryanthe Malliaris and Saharon Shelah submitted to arxiv and the Journal of the AMS a paper in which they proved that $\textbf{p}=\textbf{t}$ .

Short context: Cantor proved in 1874 that the cardinality of set of integers (denoted $\aleph_0$ ) is strinctly less than the cardinality of set of real numbers (denoted $2^{\aleph_0}$ ). Clearly, both $\textbf{p}$ and $\textbf{t}$ are at least $\aleph_0$ and no more than $2^{\aleph_0}$ . It is easy to see that $\textbf{p}\leq\textbf{t}$ , since a tower has the s.f.i.p. An old open question asks whether equality holds. The Theorem asnwers this question affirmatively.