Max Versus Max

The $2n$ consecutive integers $1, 2, \ldots, 2n$ (where $n \in \mathbb{N}^*$, $n \geqslant 2$) are randomly split into two groups $A$ and $B$ of $n$ numbers each. Let $a$ be the largest number in group $A$, let $b$ be the largest number in group $B$, and set $\xi = |a - b|$.

1. For $n = 3$, find the expected value $E(\xi)$. 2. Find the smallest constant $c$ such that $E(\xi) < c$ for every $n \geqslant 2$.