The maximum product of weights of cross-intersecting families

TL;DR

This paper establishes a new theorem for weighted cross-$t$-intersecting families, determining the maximum product of weights and sizes, with applications to various combinatorial families and generalizations.

Contribution

It introduces a novel subfamily alteration method to solve maximum product problems for weighted cross-$t$-intersecting families, extending classical results in extremal set theory.

Findings

Maximum product of weights achieved by specific families

Explicit formula for maximum size product in certain families

Generalizations to multiple families and Erdős-Ko-Rado-type results

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.