The maximum product of weights of cross-intersecting families

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 $\left(\genfrac{}{}{0px}{}{[n]}{r}\right)$ be the family of $r$-element subsets of $[n]$, and let $\left(\genfrac{}{}{0px}{}{[n]}{\le r}\right)$ 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 $\left(\genfrac{}{}{0px}{}{[n]}{\le r}\right)$ that contain $[t]$. We show that if $g:\left(\genfrac{}{}{0px}{}{[m]}{\le r}\right)\to {\mathbb{R}}^{+}$ and $h:{[n]\choose\leq s}\rightarrow$\mathbb{R}$^+$arefunctionsthatobeycertainconditions,$$\mathcal{A}$\subseteq{[m]\choose\leq r}$,$$\mathcal{B}$\subseteq{[n]\choose\leq s}$,and$$\mathcal{A}$$and$$\mathcal{B}$$arecross-$t$-intersecting, then $$\sum _{A\in \mathcal{A}}g(A)\sum _{B\in \mathcal{B}}h(B)\le \sum _{C\in {\mathcal{F}}_{m,r,t}}g(C)\sum _{D\in {\mathcal{F}}_{n,s,t}}h(D),$$ 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 $\mathcal{A}\subseteq \left(\genfrac{}{}{0px}{}{[m]}{r}\right)$ and $\mathcal{B}\subseteq \left(\genfrac{}{}{0px}{}{[n]}{s}\right)$ is $\left(\genfrac{}{}{0px}{}{m-t}{r-t}\right)\left(\genfrac{}{}{0px}{}{n-t}{s-t}\right)$ for $\min \{m,n\}\ge n_0(r,s,t)$, 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 $k\ge 2$ cross-$t$-intersecting families, and Erdos-Ko-Rado-type results.