Erdős-Ko-Rado theorem for vector spaces over residue class rings

Abstract

Let $h={\prod }_{i=1}^t{p}_i^{s_i}$ be its decomposition into a product of powers of distinct primes, and ${\mathbb{Z}}_h$ be the residue class ring modulo $h$. Let ${\mathbb{Z}}_h^n$ be the $n$-dimensional row vector space over ${\mathbb{Z}}_h$. A generalized Grassmann graph for ${\mathbb{Z}}_h^n$, denoted by $G_r(m,n,{\mathbb{Z}}_h)$ ($G_r$ for short), has all $m$-subspaces of ${\mathbb{Z}}_h^n$ as its vertices, and two distinct vertices are adjacent if their intersection is of dimension $>m-r$, where $2\le r\le m+1\le n$. In this paper, we determine the clique number and geometric structures of maximum cliques of $G_r$. As a result, we obtain the Erdős-Ko-Rado theorem for ${\mathbb{Z}}_h^n$.

18 pages