Introduction:
The Collegium System is a method of appointment and transfer of judges in the higher judiciary of India. It was introduced as a result of judicial interpretation of the Constitution by the Supreme Court in a series of cases. The Third Judges Case, decided in 1998, played a significant role in establishing the Collegium System.

Statement 1: The Collegium System was introduced in the Third Judges Case (1998).
The statement is correct.
Explanation:
The Collegium System was indeed introduced in the Third Judges Case, which is officially known as the Supreme Court Advocates-on-Record Association v. Union of India case. In this case, the Supreme Court held that the process of appointing judges to the higher judiciary must involve the judiciary itself and that the opinion of the Chief Justice of India (CJI) should have primacy in the appointment process.

Statement 2: The government plays no role regarding the appointment of judges in a High Court Collegium.
The statement is not entirely correct.
Explanation:
The government does have a role in the appointment process of judges in a High Court Collegium. While the Collegium System gives the judiciary a significant say in the appointment of judges, the government's role cannot be completely disregarded. The CJI, along with a group of senior judges, typically forms the Collegium at the High Court level.

The government's role comes into play when the Collegium recommends candidates for appointment. The government has the authority to either accept or reject the recommendations made by the Collegium. However, the government is bound by the principle of "consultation" with the judiciary, which means it cannot unilaterally reject the recommendations without providing valid reasons.

In practice, the government and the Collegium engage in a consultative process to reach a consensus on the appointment of judges. The government may seek clarifications or express reservations about certain candidates, and the Collegium may take these into consideration while making its final recommendations.

Conclusion:
In summary, the Collegium System was indeed introduced in the Third Judges Case (1998). While the government does play a role in the appointment of judges in a High Court Collegium, it is bound by the principle of consultation with the judiciary. The process involves a consultative process between the government and the Collegium to ensure the appointment of competent and suitable judges to the higher judiciary.