IMC 1997 · Problem 6

Suppose that $F$ is a family of finite subsets of $\mathbb{N}$ and for any two sets $A, B \in F$ we have $A \cap B \ne \emptyset$.

a) Is it true that there is a finite subset $Y$ of $\mathbb{N}$ such that for any $A, B \in F$ we have $A \cap B \cap Y \ne \emptyset$?

b) Is the statement a) true if we suppose in addition that all of the members of $F$ have the same size?

Justify your answers.