A cross-intersection theorem for subsets of a set

Two families $\mathcal{A}$ and $\mathcal{B}$ of sets are said to be cross-intersecting if each member of $\mathcal{A}$ intersects each member of $\mathcal{B}$. For any two integers $n$ and $k$ with $0 \leq k \leq n$, let ${[n] \choose \leq k}$ denote the family of all subsets of $\{1, \dots, n\}$ of size at most $k$. We show that if $\mathcal{A} \subseteq {[m] \choose \leq r}$, $\mathcal{B} \subseteq {[n] \choose \leq s}$, and $\mathcal{A}$ and $\mathcal{B}$ are cross-intersecting, then $$|\mathcal{A}||\mathcal{B}| \leq \sum_{i=0}^r {m-1 \choose i-1} \sum_{j=0}^s {n-1 \choose j-1},$$ and equality holds if $\mathcal{A} = \{A \in {[m] \choose \leq r} \colon 1 \in A\}$ and $\mathcal{B} = \{B \in {[n] \choose \leq s} \colon 1 \in B\}$. Also, we generalise this to any number of such cross-intersecting families.Comment: 12 pages, submitted. arXiv admin note: text overlap with arXiv:1212.695

Cross-intersecting sub-families of hereditary families

Families $\mathcal{A}_1, \mathcal{A}_2, ..., \mathcal{A}_k$ of sets are said to be \emph{cross-intersecting} if for any $i$ and $j$ in $\{1, 2, ..., k\}$ with $i \neq j$, any set in $\mathcal{A}_i$ intersects any set in $\mathcal{A}_j$. For a finite set $X$, let $2^X$ denote the \emph{power set of $X$} (the family of all subsets of $X$). A family $\mathcal{H}$ is said to be \emph{hereditary} if all subsets of any set in $\mathcal{H}$ are in $\mathcal{H}$; so $\mathcal{H}$ is hereditary if and only if it is a union of power sets. We conjecture that for any non-empty hereditary sub-family $\mathcal{H} \neq \{\emptyset\}$ of $2^X$ and any $k \geq |X|+1$, both the sum and product of sizes of $k$ cross-intersecting sub-families $\mathcal{A}_1, \mathcal{A}_2, ..., \mathcal{A}_k$ (not necessarily distinct or non-empty) of $\mathcal{H}$ are maxima if $\mathcal{A}_1 = \mathcal{A}_2 = ... = \mathcal{A}_k = \mathcal{S}$ for some largest \emph{star $\mathcal{S}$ of $\mathcal{H}$} (a sub-family of $\mathcal{H}$ whose sets have a common element). We prove this for the case when $\mathcal{H}$ is \emph{compressed with respect to an element $x$ of $X$}, and for this purpose we establish new properties of the usual \emph{compression operation}. For the product, we actually conjecture that the configuration $\mathcal{A}_1 = \mathcal{A}_2 = ... = \mathcal{A}_k = \mathcal{S}$ is optimal for any hereditary $\mathcal{H}$ and any $k \geq 2$, and we prove this for a special case too.Comment: 13 page

Strongly intersecting integer partitions

We call a sum a1+a2+• • •+ak a partition of n of length k if a1, a2, . . . , ak and n are positive integers such that a1 ≤ a2 ≤ • • • ≤ ak and n = a1 + a2 + • • • + ak. For i = 1, 2, . . . , k, we call ai the ith part of the sum a1 + a2 + • • • + ak. Let Pn,k be the set of all partitions of n of length k. We say that two partitions a1+a2+• • •+ak and b1+b2+• • •+bk strongly intersect if ai = bi for some i. We call a subset A of Pn,k strongly intersecting if every two partitions in A strongly intersect. Let Pn,k(1) be the set of all partitions in Pn,k whose first part is 1. We prove that if 2 ≤ k ≤ n, then Pn,k(1) is a largest strongly intersecting subset of Pn,k, and uniquely so if and only if k ≥ 4 or k = 3 ≤ n ̸∈ {6, 7, 8} or k = 2 ≤ n ≤ 3.peer-reviewe