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 …

# Category: Scrapbook

## Piecewise syndetic and van der Waerden

Joel Moreira has just proved that whenever the natural numbers are finitely coloured we can find and such that , and are all the same colour. He actually proves a much more general result via links to topological dynamics, but he includes a direct proof of this special case assuming only a consequence of van der Waerden’s …