What is the Royal Nonesuch and what does the Duke mean when he says, "I don't know Arkansaw!"

The Duke and King are trying to increase spectators to their theatrical productions. After poor turnout to their badly performed Shakespeare Recitation, the Duke decides that what these "Arkansaw lunkheads" want is low comedy -- or maybe something worse than low comedy.  So he and the King create the "tragedy" of "The Royal Nonesuch,"  in which the King prances around the stage naked. While the Duke was making posters to advertise the event, he wrote LADIES AND CHILDREN NOT ADMITTED on as the biggest and last line, to suggest this was to be a sexual performance of some sort.  His statement, "If that don't fetch them, I don't know Arkansaw!" meant that he was appealing to the lowest form of sexual humor that the residents of Arkansaw possessed, enticing them to purchase a ticket for admission to the play.