[Paper Review] On Cross-intersecting Sperner Families

Wiseley Wong, Eng Guan Tay|arXiv (Cornell University)|Jan 7, 2020

Limits and Structures in Graph Theory15 references4 citations

TL;DR

This paper establishes tight upper bounds on the sum of sizes of two cross-intersecting Sperner families (antichains) on an $n$-element set, proving that $|\mathscr{A}| + |\mathscr{B}| \leq \binom{n}{\lfloor n/2 \rfloor} + \binom{n}{\lceil n/2 \rceil}$, with equality if and only if the families are the full collections of $\lfloor n/2 \rfloor$-sets and $\lceil n/2 \rceil$-sets. The proof uses Sperner operations and shadow/shade techniques to characterize extremal and almost-extremal configurations.

ABSTRACT

Two sets $\mathscr{A}$ and $\mathscr{B}$ are said to be cross-intersecting if $X\cap Y eq\emptyset$ for all $X\in\mathscr{A}$ and $Y\in\mathscr{B}$. Given two cross-intersecting Sperner families (or antichains) $\mathscr{A}$ and $\mathscr{B}$ of $\mathbb{N}_n$, we prove that $|\mathscr{A}|+|\mathscr{B}|\le 2{{n}\choose{\lceil{n/2} ceil}}$ if $n$ is odd, and $|\mathscr{A}|+|\mathscr{B}|\le {{n}\choose{n/2}}+{{n}\choose{(n/2)+1}}$ if $n$ is even. Furthermore, all extremal and almost-extremal families for $\mathscr{A}$ and $\mathscr{B}$ are determined.

Motivation & Objective

• To determine the maximum possible sum of sizes of two cross-intersecting Sperner families on $\mathbb{N}_n$.
• To characterize all extremal and almost-extremal families achieving these bounds.
• To extend classical results on intersecting families to the Sperner (antichain) setting with cross-intersection constraints.
• To provide a new proof using Sperner operations, differing from prior approaches.
• To connect the extremal set-theoretic results to graph theory, particularly optimal orientations in $G$-vertex multiplications.

Proposed method

• Utilizes Sperner operations (shadows and shades) to analyze and transform families while preserving cross-intersection and antichain properties.
• Applies the Lubell-Yamamoto-Meshalkin (LYM) inequality framework and related shadow theorems to bound family sizes.
• Employs case analysis based on the sizes of families in middle levels $\lfloor n/2 \rfloor$ and $\lceil n/2 \rceil$, leveraging known inequalities on shadow growth.
• Uses double counting and extremal set theory techniques to derive contradictions for non-extremal configurations.
• Applies the Kruskal-Katona theorem implicitly via shadow size estimates to bound the number of sets in upper levels.
• Establishes uniqueness of extremal families by showing that any deviation leads to a strict decrease in the sum $|\mathscr{A}| + |\mathscr{B}|$.

Experimental results

Research questions

• RQ1 What is the maximum possible sum $|\mathscr{A}| + |\mathscr{B}|$ for two cross-intersecting Sperner families on $\mathbb{N}_n$?
• RQ2 When is this maximum sum achieved, and what is the exact structure of the extremal families?
• RQ3 What are the almost-extremal configurations, and how do they deviate from the extremal ones?
• RQ4 Can the extremal bound be strengthened into an LYM-type inequality, as suggested by the authors?
• RQ5 How do these results relate to optimal orientations in $G$-vertex multiplications of graphs?

Key findings

• For odd $n$, the maximum sum $|\mathscr{A}| + |\mathscr{B}|$ is $2\binom{n}{\lceil n/2 \rceil}$, achieved if and only if $\{\mathscr{A}, \mathscr{B}\} = \left\{ \binom{\mathbb{N}_n}{\lfloor (n+1)/2 \rfloor}, \binom{\mathbb{N}_n}{\lceil (n+1)/2 \rceil} \right\}$.
• For even $n$, the maximum sum is $\binom{n}{n/2} + \binom{n}{n/2 + 1}$, achieved uniquely when one family is the full $n/2$-sets and the other the full $(n/2+1)$-sets.
• The almost-extremal case for odd $n$ occurs only when one family is the full $\lceil n/2 \rceil$-sets and the other is a subfamily of size $\binom{n}{\lceil n/2 \rceil} - 1$.
• For even $n$, the almost-extremal sum $\binom{n}{n/2} + \binom{n}{n/2 + 1} - 1$ occurs in exactly two symmetric configurations: one family full in level $n/2$, the other almost full in level $n/2 + 1$, or vice versa.
• All extremal and almost-extremal configurations are completely characterized, with no other families achieving these bounds.
• The proof technique using Sperner operations provides a novel and self-contained derivation, distinct from prior proofs based on $t$-intersecting families.