Erdős Problem #1028 — Imbalance of Two-Colorings of Pairs

For a finite set $X\subseteq\mathbb{N}$ and a coloring $f:\mathbb{N}\times\mathbb{N}\to\{-1,1\}$, define its imbalance on $X$ by

$$\left|\sum_{x<y\in X}f(x,y)\right|.$$

Let $H(n)$ be the minimum, over all such colorings of pairs from $\{1,\ldots,n\}$, of the maximum imbalance over all subsets $X\subseteq\{1,\ldots,n\}$. Determine the asymptotic order of $H(n)$; specifically, prove or disprove that

$$H(n)=\Theta\left(n^{3/2}\right)$$

as $n\to\infty$.