What does the lady tempt Gawain with on the third day? Why does he not accept it?

The third day dawns with a description of its brilliant, wintry beauty, and the hunting dogs fall on the trail of a cunning fox, which constantly outwits and eludes the hunting party. Inside the castle, the lady enters Gawain's bedchamber while wearing a lovely and very revealing gown. She wakes him from his sorrowful slumber, as he dreads the impending day of doom at the Green Chapel. Relentless and charming as ever, she kisses him and asks if he is not promised to another lady elsewhere. Gawain denies this, and the lady begs him to leave her a token of remembrance. She offers him a valuable ring of gold, which he kindly refuses, as he has nothing to give her. The lady then offers him a green silk tunic, which he at first refuses, but then she reveals that whoever wears the green girdle cannot be killed. Aware of his impending meeting with the Green Knight, Gawain accepts the girdle, which the lady begs to keep secret.