# Friends at infinity

**Discrete Mathematics**Level 4

We call a **friendship island** (FI) a group of people such that everyone in the FI can reach anyone else in the FI by passing a note through mutual friends. If two people cannot send notes through a series of mutual friends, they must be in different FI.

At some value of \(\langle k \rangle\), \(\langle k\rangle_c\), the expected size of the largest FI becomes \(\infty\). What is the value of \(\langle k \rangle_c\)?

**Notes and assumptions**

- There are infinitely many people in the population.