Large braces to specify condition in a set

Using the amsmath package, the proper solution would be to break the long definition using multline (or multline* for no equation number):

\begin{multline}
A = \{ (x, y, z) \colon \exists \dots \forall \dots \\
    \text{(your big definition of a graph goes here)} \}
\end{multline}

If you insist on having large braces around the definition of your set you can use something like:

\begin{equation}
A = \left\{ 
  \begin{aligned}
  &(x, y, z) \colon \exists \dots \forall \dots \\ 
  &\text{(your big definition of a graph goes here)}
  \end{aligned}
\right\}
\end{equation}

In your preamble:

\usepackage{braket} % needed for \Set

In your math mode:

A = \Set{ (x,y,z) : \exists ... \forall ... (big definition of a graph)}

For one line, you can use

\left\{ (x,y,z) : bla bla \right\}

For multiple lines, for example

\begin{align*}
      & \left\{ (x,y,z) : bla bla \right. \\
      & \hspace{1cm} \left. bla bla bla \right\}
\end{align*}