On the number of disjoint subsets of a large set families

Question

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 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 ${n -1 \choose k-1}$?

Answer 1

Your first question is simply asking what is the minimum number of edges an $n$-vertex graph must have to force a matching of size $m$. This number was determined exactly by a classic result of Erdős and Gallai. 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 by Frankl and Kupavskii. See the references therein for more information.