Why do you think King Arthur allow Sir Gawain to take up the challenge?
The main reason is that King Arthur doesn't want to look weak in front of his guests. He's not just the king; he's also a brave warrior, the protector of his people. So when the Green Knight throws down a challenge he has no choice but to accept.
Though Arthur is initially hesitant—he doesn't quite know what he's letting himself in for—he knows that it's his duty to respond to any challenge made to him. And the Green Knight's challenge isn't just a challenge to Arthur as a man; it's a challenge to him as king. If Arthur lets this challenge slide, therefore, he's allowing the Green Knight to undermine the stability of his throne. In turn, that will send out the very dangerous message that any other warrior who fancies his chances can simply rock up at Camelot any time he chooses and make the king look small in front of his knights and courtiers.
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