# Fractional cross intersecting families

## TL;DR

This paper establishes tight bounds on the sizes of two families of subsets of [n] with fractional intersection conditions, generalizing classical intersecting family results and characterizing extremal cases.

## Contribution

It introduces the concept of fractional cross intersecting families and provides tight bounds and characterizations for their maximum product sizes, including special cases like half-intersecting families.

## Key findings

- Derived tight upper bounds for the product of family sizes.
- Characterized extremal families when the bounds are achieved.
- Extended classical intersection results to fractional intersection scenarios.

## Abstract

Let $\mathcal{A}=\{A_{1},...,A_{p}\}$ and $\mathcal{B}=\{B_{1},...,B_{q}\}$ be two families of subsets of $[n]$ such that for every $i\in [p]$ and $j\in [q]$, $|A_{i}\cap B_{j}|= \frac{c}{d}|B_{j}|$, where $\frac{c}{d}\in [0,1]$ is an irreducible fraction. We call such families "$\frac{c}{d}$-cross intersecting families". In this paper, we find a tight upper bound for the product $|\mathcal{A}||\mathcal{B}|$ and characterize the cases when this bound is achieved for $\frac{c}{d}=\frac{1}{2}$. Also, we find a tight upper bound on $|\mathcal{A}||\mathcal{B}|$ when $\mathcal{B}$ is $k$-uniform and characterize, for all $\frac{c}{d}$, the cases when this bound is achieved.