Title: Beyond the Erdős Matching Conjecture,
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 Erdős Matching Conjecture
and prove a generalization of the Erdős-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 Erdős Matching Conjecture, in general there may be $3$ extremal
families.