Step 1: Name the polymerisation involved.
Ethene is polymerised to polyethylene using the Ziegler-Natta catalyst, which gives controlled, linear chains.
Step 2: Recall the two parts of this catalyst.
A Ziegler-Natta system always has a transition metal halide together with an organometallic compound of aluminium.
Step 3: Identify the transition metal halide.
The standard transition metal component used is titanium tetrachloride, $TiCl_4$.
Step 4: Identify the co-catalyst.
The aluminium part is trimethyl aluminium, $Al(CH_3)_3$, which activates the titanium centre.
Step 5: Combine them.
So the catalyst mixture is $TiCl_4$ together with $Al(CH_3)_3$, matching option 4.
Step 6: Reject the other choices.
The combinations using $CH_3MgBr$ or $Si(CH_3)_4$, or just plain $Ti$, are not the recognised Ziegler-Natta system for polyethylene.
\[ \boxed{TiCl_4,\ Al(CH_3)_3} \]