Step 1: Decode the abbreviation HDP.
HDP stands for High-Density Polyethylene, the tightly packed, mostly linear form of polyethene.
Step 2: Connect linear chains to the catalyst type.
To get straight chains with almost no branching, the ethene must add in a controlled, coordinated way. That control comes from a coordination catalyst, not a random radical process.
Step 3: Name that coordination catalyst.
The classic coordination system is titanium tetrachloride with triethylaluminium, $\text{TiCl}_4 + \text{Al(C}_2\text{H}_5)_3$, known as the Ziegler-Natta catalyst.
Step 4: Rule out peroxides.
Peroxides are free-radical initiators. They create branched chains and give Low-Density Polyethylene (LDP), the opposite of what we want.
Step 5: Rule out Lindlar's catalyst.
Lindlar's catalyst ($\text{Pd/CaCO}_3$, poisoned) only partially reduces alkynes to cis-alkenes. It has nothing to do with polymerising ethene.
Step 6: Rule out magnesium oxide.
$\text{MgO}$ is a basic refractory oxide and does not polymerise ethene.
Step 7: Select the answer.
Only Ziegler-Natta fits HDP manufacture, option (1).
\[ \boxed{\text{Ziegler-Natta catalyst}} \]