On $r$-cross $t$-intersecting families for weak compositions

Abstract

Let ${\mathbb{N}}_0$ be the set of non-negative integers, and let $P(n,l)$ denote the set of all weak compositions of $n$ with $l$ parts, i.e., $P(n,l)=\{(x_1,x_2,\dots ,x_l)\in {\mathbb{N}}_0^l:x_1+x_2+\cdots +x_l=n\}$. For any element $\mathbf{u}=(u_1,u_2,\dots ,u_l)\in P(n,l)$, denote its $i$th-coordinate by $\mathbf{u}(i)$, i.e., $\mathbf{u}(i)=u_i$. Let $l=\min (l_1,l_2,\dots ,l_r)$. Families ${\mathcal{A}}_j\subseteq P(n_j,l_j)$ ($j=1,2,\dots ,r$) are said to be $r$-cross $t$-intersecting if $|\{i\in [l]:{\mathbf{u}}_1(i)={\mathbf{u}}_2(i)=\cdots ={\mathbf{u}}_r(i)\}|\ge t$ for all ${\mathbf{u}}_j\in {\mathcal{A}}_j$. Suppose that $l\ge t+2$. We prove that there exists a constant $n_0=n_0(l_1,l_2,\dots ,l_r,t)$ depending only on $l_j$'s and $t$, such that for all $n_j\ge n_0$, if the families ${\mathcal{A}}_j\subseteq P(n_j,l_j)$ ($j=1,2,\dots ,r$) are $r$-cross $t$-intersecting, then \begin{equation} \prod_{j=1}^r \vert \mathcal{A}_j \vert\leq \prod_{j=1}^r {n_j+l_j-t-1 \choose l_j-t-1}.\notag \end{equation} Moreover, equality holds if and only if there is a $t$-set $T$ of $\{1,2,\dots ,l\}$ such that ${\mathcal{A}}_j=\{\mathbf{u}\in P(n_j,l_j):\mathbf{u}(i)=0\mathrm{for}\mathrm{all}i\in T\}$ for $j=1,2,\dots ,r$.

9 pages. arXiv admin note: text overlap with arXiv:1311.1592

References in corpus (2)

• Shadows and intersections: stability and new proofs
• Erdos-Ko-Rado theorems for simplicial complexes