There are infinitely many monotone games over $L_5$

TL;DR

This paper proves that when the set of atomic outcomes in monotone combinatorial games over a poset has five or more elements, there are infinitely many distinct game values, completing the classification for all sizes.

Contribution

It establishes that for atom posets with five or more elements, the set of monotone game values is infinite, resolving a previously open case.

Findings

Infinite monotone game values when atom poset has ≥5 elements

Finite values for atom posets with ≤4 elements

Completes classification of monotone game values based on atom poset size

Abstract

A notion of combinatorial game over a partially ordered set of atomic outcomes was recently introduced by Selinger. These games are appropriate for describing the value of positions in Hex and other monotone set coloring games. It is already known that there are infinitely many distinct monotone game values when the poset of atoms is not linearly ordered, and that there are only finitely many such values when the poset of atoms is linearly ordered with 4 or fewer elements. In this short paper, we settle the remaining case: when the atom poset has 5 or more elements, there are infinitely many distinct monotone values.