Final answer:
Aegeus demands a pledge of loyalty from Medea to provide her with shelter in his country.
Step-by-step explanation:
The requirement that Aegeus demands to give Medea shelter is a pledge of loyalty. In the story, Aegeus asks Medea to swear by the gods that she will provide him with certain benefits, and in return, he will give her refuge in his land. This moment in the narrative underlines the importance of oaths in Greek mythology and the political uses of sanctuary and protection promises. Medea, known for her cunning and magical powers, fulfills this requirement as she seeks safety and the opportunity to execute her plans of revenge against her ex-husband, Jason, who abandoned her.