The maximum sum and maximum product of sizes of cross-intersecting families

TL;DR

This paper investigates the maximum sum and product of sizes of cross-$t$-intersecting families within a finite family of sets, establishing conditions for optimality and exploring specific cases including power sets.

Contribution

It proves that for large enough number of families, the maximum sum and product are achieved when all families are identical largest $t$-intersecting sub-families, and analyzes the minimal such number.

Findings

Maximum sum and product are achieved by identical largest $t$-intersecting sub-families for sufficiently large k.

Identifies the minimal k (kappa) for which the optimal configuration holds.

Provides solutions for specific families, including power sets, and discusses cases where k < kappa.

Abstract

We say that a set $A$ \emph{$t$-intersects} a set $B$ if $A$ and $B$ have at least $t$ common elements. A family $\mathcal{A}$ of sets is said to be \emph{$t$-intersecting} if each set in $\mathcal{A}$ $t$-intersects any other set in $\mathcal{A}$. Families $\mathcal{A}_1, \mathcal{A}_2, ..., \mathcal{A}_k$ are said to be \emph{cross-$t$-intersecting} if for any $i$ and $j$ in $\{1, 2, ..., k\}$ with $i \neq j$, any set in $\mathcal{A}_i$ $t$-intersects any set in $\mathcal{A}_j$. We prove that for any finite family $\mathcal{F}$ that has at least one set of size at least $t$, there exists an integer $\kappa \leq |\mathcal{F}|$ such that for any $k \geq \kappa$, both the sum and the product of sizes of any $k$ cross-$t$-intersecting sub-families $\mathcal{A}_1, ..., \mathcal{A}_k$ (not necessarily distinct or non-empty) of $\mathcal{F}$ are maxima if $\mathcal{A}_1 = ... = \mathcal{A}_k = \mathcal{L}$ for some largest $t$-intersecting sub-family $\mathcal{L}$ of $\mathcal{F}$. We then study the smallest possible value of $\kappa$ and investigate the case $k < \kappa$; this includes a cross-intersection result for straight lines that demonstrates that it is possible to have $\mathcal{F}$ and $\kappa$ such that for any $k < \kappa$, the configuration $\mathcal{A}_1 = ... = \mathcal{A}_k = \mathcal{L}$ is neither optimal for the sum nor optimal for the product. We also outline solutions for various important families $\mathcal{F}$, and we provide solutions for the case when $\mathcal{F}$ is a power set.