Description
One of the first results in extremal set systems was [Sperner, 1928]
asserting that if no two members of a system $\mathcal{F}$ of subsets of
an n element underlying set contain each other, then $\mathcal{F}$ has
at most $\binom{n}{\lfloor n/2 \rfloor}$ members. We call a set system
$\mathcal{F}$ a $d$-union, if the union of any members of $\mathcal{F}$
has at most $d$ elements. Millner's result [1968], recognized today as
classic, claims that if a Sperner system $\mathcal{F}$ is a $d$-union,
then is has at most $\binom{n}{\lfloor d/2 \rfloor}$ members. In 2014,
Matas Sileikis posed a question, similar to Millner's, to bound the size
of $\mathcal{F}$ if, instead of the union, we put restrictions on the
symmetric differences. (The maximum size of symmetric differences is
called the diameter of $\mathcal{F}$.)
The main result in the talk will be the answer for this question: Péter
Frankl has shown that for any fixed $d$ there is a threshold $n_0(d)$
such that if $n > n_0(d)$, then the maximal size of a Sperner system of
diameter $d$ over an $n$ element underlying set is also
$\binom{n}{\lfloor d/2 \rfloor}$; moreover, equality holds only if each
member of $\mathcal{F}$ is of size $\lfloor d/2 \rfloor$, or each
member is of size $\lceil d/2 \rceil$.