Diversity and intersecting theorems for weak compositions

Abstract

Let $N_0$ be the set of non-negative integers, and let $P(n,k)$ denote the set of all weak compositions of n with k parts, i.e., $P(n,k)=\left\{\right(x_1,x_2,\dots ,x_k)\in N_0^k:x_1+x_2+\cdots +x_k=n\}$. For any element $u=(u_1,u_2,\dots ,u_k)\in P(n,k)$, denote its ith-coordinate by $u\left(i\right)$, i.e., $u\left(i\right)=u_i$. A family $A\subseteq P(n,k)$ is said to be t-intersecting if $\left|\right\{i:u\left(i\right)=v\left(i\right)\left\}\right|\ge t$ for all $u,v\in A$. In this paper, we consider the diversity and other intersecting theorems for weak compositions.