Algebra Seminar talk

Peter Holy
Infinite cut and choose games

We consider the following two player game of infinite length: We are given a starting set X (for example, X is the set of all real numbers say), and the players go by the names “Cut” and “Choose”. They take turns making moves, and in each step, Cut partitions a given set into two disjoint pieces, starting from the set X in their first move, and then Choose gets to pick one of the pieces, which is then partitioned into two pieces by Cut in their next move etc. In the end, Choose wins in case the intersection of all of their choices has at least two (distinct) elements.

We will investigate some of the properties of this game — in particular, we will discuss when it is possible for one of the players to have a strategy for winning the game. We will then continue to discuss some variations of this game and their relevance to set theory — many central set theoretic notions, such as certain large cardinal properties, notions of distributivity, precipitousness and strategic closure turned out to be closely connected and often equivalent to the (non-)existence of winning strategies in certain cut and choose games. Most of this talk should be accessible to a general math audience.

This is joint work with Philipp Schlicht, Christopher Turner and Philip Welch (all University of Bristol).