Games and Complexes II: Weight Games and Kruskal-Katona Type Bounds

TL;DR

This paper explores weight games, a subclass of placement games, establishing upper bounds on the number of game positions with a given number of pieces, inspired by Kruskal-Katona bounds in combinatorics.

Contribution

It introduces new upper bounds on the face counts of simplicial complexes associated with weight games, extending combinatorial bounds to this class of games.

Findings

Derived upper bounds on the number of positions with i pieces in weight games.

Established connections between placement games and simplicial complexes.

Extended Kruskal-Katona type bounds to weight games.

Abstract

A strong placement game $G$ played on a board $B$ is equivalent to a simplicial complex $\Delta_{G,B}$. We look at weight games, a subclass of strong placement games, and introduce upper bounds on the number of positions with $i$ pieces in $G$, or equivalently the number of faces with $i$ vertices in $\Delta_{G,B}$, which are reminiscent of the Kruskal-Katona bounds.