Let $[n] := \{1,\dots,n\}$, for some large integer $n$, and let $\mathcal{F}$ be a family of 2-element subsets of $[n]$.


The famous [Erdös-Ko-Rado (EKR) theorem](https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Ko%E2%80%93Rado_theorem) says that if $|\mathcal{F}| > {n - 1 \choose 1} = n-1$, then $\mathcal{F}$ must contain (at least) two disjoint subsets.


**Question:**


* Assuming that $|\mathcal{F}|$ is significantly larger than the bound required by the EKR theorem, is there a result that gives a lower bound on the number of *pairwise* disjoint 2-element subsets that $\mathcal{F}$ must contain?
* More generally, does there exist such a lower bound on the number of disjoint $k$-subsets if we consider families of $k$-element subsets that are much larger than ${n -1 \choose k-1}$?



On the number of disjoint subsets of a large set families
Your first question is simply asking what is the minimum number of edges an $n$-vertex graph must have to force a [matching](https://en.wikipedia.org/wiki/Matching_(graph_theory)) of size $m$. This number was determined exactly by a classic result of [Erdős and Gallai](https://citeseerx.ist.psu.edu/viewdoc/summary;jsessionid=1D0279A99A414F91F8003CD4304965E9?doi=10.1.1.210.7468). They proved that if the maximum size of a matching of an $n$-vertex graph $G$ is $m$, then $G$ has at most $\max\{\binom{2m+1}{2}, \binom{m}{2}+(n-m)m\}$ edges. Moreover, this bound is tight for all $n,m \geq 1$.


Your second question was asked by Erdős in 1965 and has received considerable attention. I believe the state of the art is the paper [The Erdős Matching Conjecture and concentration inequalities](https://arxiv.org/abs/1806.08855) by Frankl and Kupavskii. See the references therein for more information.


