Previously i always thought that mihawk became a shichibukai after shanks lost his arm.
His vivre card established him as famous before the great pirate era, and it establishes that he went around challenging people until there were none left. Then it says he became a shichibukai... oda can either have mihawk and shanks fight during the time that mihawk was coming up in the world or have them fight during the time that mihawk became a shichibukai... Personally now i feel like the latter applies to mihawk he became famous before the great pirate era,crushed marines as the marine hunter and other pirates as well,became a shichibukai to travel freely, him and shanks legendary duel and after that shanks loses his arm so we have bored mihawk in the current story who views shanks as a hasbeen. I do think the great betrayal is shanks losing his arm unless his grudge against the marines and the great betrayal are synonymous.