A generalization of repetition threshold

Brandenburg and (implicitly) Dejean introduced the concept of repetition threshold: the smallest real number alpha such that there exists an infinite word over a k-letter alphabet that avoids beta-powers for all beta > alpha. We generalize this concept to include the lengths of the avoided words. We give some conjectures supported by numerical evidence and prove three of these conjectures. As a consequence of one of our results, we show that the pattern ABCBABC is 2-avoidable. This resolves a question left open in Cassaigne's thesis.