1. High-density polyethylene (HDPE) is produced via Ziegler-Natta catalysis, which uses a combination of triethylaluminium ($(\text{C}_2\text{H}_5)_3\text{Al}$) and titanium tetrachloride ($\text{TiCl}_4$).
2. This catalyst enables stereospecific polymerization of ethylene, forming linear, high-density polymers.
3. Option (2) benzoyl peroxide is used for free radical polymerization (e.g., polystyrene). Option (3) persulphate initiates low-density polyethylene. Option (4) is used in polyester synthesis.
4. Thus, the answer is (1) $(\text{C}_2\text{H}_5)_3\text{Al}/\text{TiCl}_4$.