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.