Disjoint Cross Intersecting Families

TL;DR

This paper investigates the structure of disjoint cross intersecting families of r-subsets, providing new bounds on their combined size and demonstrating limitations of compression techniques in certain cases.

Contribution

It introduces a novel technique to bound the sum of sizes of disjoint cross intersecting families, overcoming limitations of traditional compression methods.

Findings

Established an upper bound for the sum of sizes of disjoint cross intersecting families.

Provided a counterexample showing compression is not always applicable.

Proved the bound is asymptotically sharp.

Abstract

For positive integers $n$ and $r$ such that $r \leq \lfloor n/2\rfloor$, let $X$ be a set of $n$ elements and let $\binom{X}{r}$ be the family of all $r$-subsets of $X$. Two sub-families $\mathcal{A}$ and $\mathcal{B}$ of $\binom{X}{r}$ are called cross intersecting if $A \cap B \neq \emptyset$ for all $A \in \mathcal{A}$ and $B \in \mathcal{B}$. One of main tools in the study of extremal set theory, and cross intersecting families in particular, is compression operation. In this paper, we give an example of cross intersecting families $\mathcal{A}$ and $\mathcal{B}$ that the compression operation is not applicable when $\mathcal{A}$ and $\mathcal{B}$ are disjoint. We develop new technique to prove that, for disjoint cross intersecting families $\mathcal{A}$ and $\mathcal{B}$ of $\binom{X}{r}$, $|\mathcal{A}| + |\mathcal{B}| \leq \binom{n}{r} - \binom{l}{p}$ where $n = 2r + l$ and $p = min\{r, \lceil \frac{l}{2}\rceil\}$. This bound is asymptotically sharp.