Abstract
Two families $\mathcal{A}$ and $\mathcal{B}$ of sets are said to be cross-intersecting if each set in $\mathcal{A}$ intersects each set in $\mathcal{B}$. For any two integers $n$ and $k$ with $1 \leq k \leq n$, let ${[n] \choose \leq k}$ denote the family of subsets of $\{1, \dots, n\}$ of size at most $k$, and let $\mathcal{S}_{n,k}$ denote the family of sets in ${[n] \choose \leq k}$ that contain $1$. The author recently showed 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 $|\mathcal{A}||\mathcal{B}| \leq \mathcal{S}_{m,r}||\mathcal{S}_{n,s}|$. We prove a version of this result for the more general setting of \emph{weighted} sets. 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-intersecting, then $$\sum_{A \in \mathcal{A}} g(A) \sum_{B \in \mathcal{B}} h(B) \leq \sum_{C \in \mathcal{S}_{m,r}} g(C) \sum_{D \in \mathcal{S}_{n,s}} h(D).$$The bound is attained by taking $\mathcal{A} = \mathcal{S}_{m,r}$ and $\mathcal{B} = \mathcal{S}_{n,s}$. We also show that this result yields new sharp bounds for the product of sizes of cross-intersecting families of integer sequences and of cross-intersecting families of multisets.
Highlights
Unless otherwise stated, we shall use small letters such as x to denote elements of a set or non-negative integers or functions, capital letters such as X to denote sets, and calligraphic letters such as F to denote families
We prove a version of this result for the more general setting of weighted sets
The bound is attained by taking A = Sm,r and B = Sn,s. We show that this result yields new sharp bounds for the product of sizes of cross-intersecting families of integer sequences and of cross-intersecting families of multisets
Summary
We shall use small letters such as x to denote elements of a set or non-negative integers or functions, capital letters such as X to denote sets, and calligraphic letters such as F to denote families (i.e. sets whose elements are sets themselves). It is natural to consider the problem of maximizing the sum or the product of sizes of k cross-intersecting subfamilies (not necessarily distinct or non-empty) of a given family F. We mention that one of the central problems in extremal set theory is a conjecture of Chvatal [22] that claims that at least one of the largest interesting subfamilies of any hereditary family H is a star of H; a similar conjecture for levels of H is made and partially solved in [15], and generalizes [37, Conjecture 7]. The critical part of the proof is the observation that if we assume that this happens for both pairs, we can construct a new pair of cross-intersecting families for which the product of weights is larger than that for A and B (unless we have the trivial case m = n = 2), contradicting the initial assumption
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