Category: Scrapbook

The Namer-Claimer game

Published by babarber on

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 …

Nowhere zero 6-flows

Published by babarber on

A flow on a graph is an assignment to each edge of of a direction and a non-negative integer (the flow in that edge) such that the flows into and out of each vertex agree.  A flow is nowhere zero if every edge is carrying a positive flow and (confusingly) it is a -flow if the flows …

Ultrafilter quantifiers and Hindman’s theorem

Published by babarber on

A filter on is a consistent notion of largeness for subsets of . “Largeness” has the following properties. if is large and then is large if and are large then is large the empty set is not large At most one of and is large; an ultrafilter is a filter which always has an opinion …

Piecewise syndetic and van der Waerden

Published by babarber on

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 …