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Tyrion is in a "sky cell," one of the Aerie's dungeon cells that has an open wall over a very far fall. Yikes.
He also has a cruel jailor named Mord (or if you prefer the old spelling, like our author, "gaoler" –same thing).
Tyrion gets a message to Lysa that he's ready to confess. But when Lysa sees him in full court, he maintains his innocence regarding Jon Arryn's death and the attempt on Bran's life. (He's only guilty of drinking too much and sleeping around.)
Tyrion demands trial by combat, where two people will fight and whoever wins will be proven innocent. Alternately, the people involved in the trial can name champions to fight for them. So Tyrion and Lysa both name champions.
Lysa makes Ser Vardis Egen fight for her, thinking it will be easy to kill the dwarf (since he has no champions). But Bronn steps forward to champion Tyrion. It's kind of like hiring a high-powered lawyer, so maybe their system of justice isn't so different from ours after all.