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

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.