# Beyond the Erd\H{o}s Matching Conjecture

## TL;DR

This paper extends classical combinatorial theorems by determining maximum family sizes under generalized union constraints, generalizing the Erdős Matching Conjecture and Erdős-Ko-Rado theorem, with specific insights for small set sizes.

## Contribution

It generalizes the Erdős Matching Conjecture and Erdős-Ko-Rado theorem to broader union properties, providing new bounds and characterizations for extremal families.

## Key findings

- Determined maximum sizes of $U(s,q)$ families for large $n$.
- Proved a generalized Erdős-Ko-Rado theorem for $n > s^2k$.
- Identified multiple extremal families for the case $k=3$.

## Abstract

A family $\mathcal F\subset {[n]\choose k}$ is $U(s,q)$ of for any $F_1,\ldots, F_s\in \mathcal F$ we have $|F_1\cup\ldots\cup F_s|\le q$. This notion generalizes the property of a family to be $t$-intersecting and to have matching number smaller than $s$. In this paper, we find the maximum $|\mathcal F|$ for $\mathcal F$ that are $U(s,q)$, provided $n>C(s,q)k$ with moderate $C(s,q)$. In particular, we generalize the result of the first author on the Erd\H{o}s Matching Conjecture and prove a generalization of the Erd\H{o}s-Ko-Rado theorem, which states that for $n> s^2k$ the largest family $\mathcal F\subset {[n]\choose k}$ with property $U(s,s(k-1)+1)$ is the star and is in particular intersecting. (Conversely, it is easy to see that any intersecting family in ${[n]\choose k}$ is $U(s,s(k-1)+1)$.) We investigate the case $k=3$ more thoroughly, showing that, unlike in the case of the Erd\H{o}s Matching Conjecture, in general there may be $3$ extremal families.