The maximum product of weights of cross-intersecting families

Abstract

Two families $\mathcal{A}$ and $\mathcal{B}$ of sets are said to be cross-$t$-intersecting if each set in $\mathcal{A}$ intersects each set in $\mathcal{B}$ in at least $t$ elements. An active problem in extremal set theory is to determine the maximum product of sizes of cross-$t$-intersecting subfamilies of a given family. We prove a cross-$t$-intersection theorem for weighted subsets of a set by means of a new subfamily alteration method, and use the result to provide solutions for three natural families. For $r\in[n]=\{1,2,\dots,n\}$, let ${[n]\choose r}$ be the family of $r$-element subsets of $[n]$, and let ${[n]\choose\leq r}$ be the family of subsets of $[n]$ that have at most $r$ elements. Let $\mathcal{F}_{n,r,t}$ be the family of sets in ${[n]\choose\leq r}$ that contain $[t]$. We show that if $g:{[m]\choose\leq r}\rightarrow\mathbb{R}^+$ and $h:{[n]\choose\leq s}\rightarrow\mathbb{R}^+$ are functions that obey certain conditions, $\mathcal{A}\subseteq{[m]\choose\leq r}$, $\mathcal{B}\subseteq{[n]\choose\leq s}$, and $\mathcal{A}$ and $\mathcal{B}$ are cross-$t$-intersecting, then \[\sum_{A\in\mathcal{A}}g(A)\sum_{B\in\mathcal{B}}h(B)\leq\sum_{C\in\mathcal{F}_{m,r,t}}g(C)\sum_{D\in\mathcal{F}_{n,s,t}}h(D),\] and equality holds if $\mathcal{A}=\mathcal{F}_{m,r,t}$ and $\mathcal{B}=\mathcal{F}_{n,s,t}$. We prove this in a more general setting and characterise the cases of equality. We use the result to show that the maximum product of sizes of two cross-$t$-intersecting families $\mathcal{A}\subseteq{[m]\choose r}$ and $\mathcal{B}\subseteq{[n]\choose s}$ is ${m-t\choose r-t}{n-t\choose s-t}$ for $\min\{m,n\}\geq n_0(r,s,t)$, where $n_0(r,s,t)$ is close to best possible. We obtain analogous results for families of integer sequences and for families of multisets. The results yield generalisations for $k\geq2$ cross-$t$-intersecting families, and Erdos-Ko-Rado-type results.