# Sharp results concerning disjoint cross-intersecting families

## TL;DR

This paper establishes sharp bounds on the maximum sizes of disjoint cross-intersecting families of k-subsets from an n-element set, revealing precise conditions under which these bounds hold.

## Contribution

It provides exact formulas and conditions for the maximum sizes of disjoint cross-intersecting families, extending previous results with sharp thresholds.

## Key findings

- f(n,k) = floor(1/2 * binomial(n-1, k-1)) for n > c k^2
- f*(n,k) = floor(1/2 * (binomial(n-1, k-1) - binomial(n-2k, k-1))) + 1 for n ≥ k^3 and k ≥ 5
- Sharp thresholds for n relative to k for the bounds to hold

## Abstract

For an $n$-element set $X$ let $\binom{X}{k}$ be the collection of all its $k$-subsets. Two families of sets $\mathcal A$ and $\mathcal B$ are called cross-intersecting if $A\cap B \neq \emptyset$ holds for all $A\in\mathcal A$, $B\in\mathcal B$. Let $f(n,k)$ denote the maximum of $\min\{|\mathcal A|, |\mathcal B|\}$ where the maximum is taken over all pairs of {\em disjoint}, cross-intersecting families $\mathcal A, \mathcal B\subset\binom{[n]}{k}$. Let $c=\log_2e$. We prove that $f(n,k)=\left\lfloor\frac12\binom{n-1}{k-1}\right\rfloor$ essentially iff $n>ck^2$ (cf. Theorem~1.4 for the exact statement). Let $f^*(n,k)$ denote the same maximum under the additional restriction that the intersection of all members of both $\mathcal A$ and $\mathcal B$ are empty. For $k\ge5$ and $n\ge k^3$ we show that $f^*(n,k)=\left\lfloor\frac12\left(\binom{n-1}{k-1}-\binom{n-2k}{k-1}\right)\right\rfloor+1$ and the restriction on $n$ is essentially sharp (cf. Theorem~5.4).