The Namer-Claimer game

Consider the following game played on the integers from to .  In each round Namer names a forbidden distance , then Claimer claims a subset of that does not contain any two integers at distance .  After finitely many rounds, Claimer will have claimed sets that cover the whole of , at which point the game ends.  How many rounds will there be with best play?

I've run this question out at a number of workshops and open problems sessions, and haven't yet heard back about a success.  I'll explain the known upper and lower bounds below the fold, but encourage you to spend a few minutes thinking about it before taking a look.

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