Counting Ordered Pairs

Question

Let $A$ be a finite set with $n \geq 4$ elements and let $\rho$ be an equivalence relation on $A$. Suppose that there are exactly $4$ equivalence classes, $C_1$, $C_2$, $C_3$, $C_4$. Moreover we know that $|C_1| = |C_2| = 1$. Let $a\in A$ be a fixed element that we know is in $C_3$. What is the maximum number of ordered pairs of $(x, y) \in \rho$ in which a can occur (meaning $a = x$ or $a = y$ or $a = x = y$)?

Answer 1

Hint: You are correct that to maximize the number of pairs you should have $|C_4|=1$, which gives $|C_3|=n-3$ But when you add the third element to $C_3$ you add more than two ordered pairs. $c$ can be paired with each of $a$ and $b$ in either order, so you add four pairs. Can you determine what happens for larger $n$?