  For a set $A\subseteq \mathbb{N}$ and $n\in \mathbb{N}$, let $R_A(n)$ denote
the number of ordered pairs $(a,a')\in A\times A$ such that $a+a'=n$. The
celebrated Erd\H{o}s-Tur\'{a}n conjecture says that, if $R_A(n)\ge 1$ for all
sufficiently large integers $n$, then the representation function $R_A(n)$
cannot be bounded. For any positive integer $m$, Ruzsa's number $R_m$ is
defined to be the least positive integer $r$ such that there exists a set
$A\subseteq \mathbb{Z}_m$ with $1\le R_A(n)\le r$ for all $n\in \mathbb{Z}_m$.
In 2008, Chen proved that $R_{m}\le 288$ for all positive integers $m$. In this
paper, we prove that $R_m\ge 6$ for all integers $m\ge 36$. We also determine
all values of $R_m$ when $m\le 35$.
