Beyond Hilton-Milner Theorem for an Intersecting Family?

Hilton-Milner tells us about an intersecting family of $k$-size subsets $\mathcal{F}$ from $\binom{[n]}{k}$: If $|\mathcal{F}| > \binom{n-1}{k-1}-\binom{n-k-1}{k-1}+1$, then all elements of $\mathcal{F}$ share at least 1 element of [n].

The theorem classifies the non-trivial $\mathcal{F}$ if equality occurs i.e. $|\mathcal{F}| = \binom{n-1}{k-1}-\binom{n-k-1}{k-1}+1$.

I was wondering if there are known results/literature to see on classifications with $|\mathcal{F}| < \binom{n-1}{k-1}-\binom{n-k-1}{k-1}+1$, particularly values very close to equality?

I am currently looking at the case of $n=7$, and $k=3$ to get an idea of this (i.e. what can be said about an intersecting family of size 12 < $\binom{6}{2}-\binom{3}{2}+1?)$.

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Does the question refer somehow to the Hilton-Milnor theorem (named after Peter Hilton and John Milnor) on homotopy groups of wedges of spheres? – Alex Suciu Apr 18 '12 at 22:30
No, this is a set theory result with Anthony Hilton and Eric Milner. – Mike Apr 19 '12 at 17:28
The set-theory tag seems misplaced here, since the topic is really finite combinatorics. – Joel David Hamkins Apr 17 '13 at 2:56

There are at least three types of results that spring to mind. One is the Ahlswede-Khachatrian theorem ("the complete intersection theorem"), which for each $n$ and $k < n/2$ will give you tight upper bounds for $t$-intersecting families (families in which every two sets intersect in at least $t$ points). For example, if $k \leq n/3 + 1$ then a $2$-intersecting family has size at most $\binom{n-2}{k-2}$, and so if $|\mathcal{F}| > \binom{n-2}{k-2}$ then there are two sets $S,T \in \mathcal{F}$ such that $|S \cap T| = 1$. They also have a matching Hilton-Milner type result, in which the family is required to have empty intersection ("the complete non-trivial intersection theorem").
A second one is stability results. Such results show that if $|\mathcal{F}| = (1-\epsilon)\binom{n-1}{k-1}$ then there is an element belonging to a $1-O(\epsilon)$ fraction of the sets. One such result is due to Friedgut ("on the measure of intersecting families, uniqueness and stability"), another is due to Keevash ("shadows and intersections: stability and new proofs").
A third one is due to Dinur and Friedgut ("intersecting families are essentially contained in juntas"). For $k = cn$, they show that any intersecting family essentially depends on $O(1)$ points, where the constant depends on $c$. For $k = o(n)$, they can show that any intersecting family has an element common to all but $C \binom{n-2}{k-2}$ of the sets.