This is a comment about shattering.

Let $U$ be a finite set. If $S\subseteq [n]$ has cardinality $d$, then
call $U^n$ an $n$-dimensional cube and
call $U^S$ the $d$-dimensional face indexed by $S$. A subset
$A\subseteq U^n$ **shatters** the face $U^S$ if the projection of
$U^n$ onto $U^S$ maps $A$ onto $U^S$.

The Shattering Lemma, due to Vapnik-Chervonenkis and also to
Perles–Sauer–Shelah, concerns the case $U=\{0,1\}$.
It is the statement that the **maximal** size of
a subset $A\subseteq \{0,1\}^n$ that shatters **no**
$d$-dimensional face is $\sum_{k<d} \binom{n}{k}$.
An explicit nonshattering set of size
$\sum_{k<d} \binom{n}{k}$ is known, namely
the set of all tuples in $\{0,1\}^n$ with fewer than
$d$ ones.

A dual combinatorial
problem came up in a paper I wrote with Emil Kiss
and Agnes Szendrei. We wanted to know:
what is the **minimal** size of a subset $A\subseteq U^n$
that shatters **every** $d$-dimensional face?
Here $U$ is allowed to be an arbitrary finite set.

We found that (if $n\geq d>1$) there is some $A\subseteq U^n$ that shatters every
$d$-dimensional face, which satisfies the bound
$$
|A|\leq \lceil d\log_b(n)+\log_b(u^d/d!)
\rceil,
$$
where $u=|U|$ and $b=u^d/(u^d-1)$. For our purposes, $u, d, b$
are fixed, but $n$ is allowed to vary.
Although this $\sim \log_b(n)$ bound may not be optimal,
we know that there is no $A\subseteq U^n$
of size $<\log_u(n)$ that shatters every $d$-dimensional
face, so $\log(n)$ is the right growth order.

The thing that might interest readers of this thread
is: we proved the existence of a fully-shattering set
of $\log(n)$-size with a probabilistic argument.
We computed the expected number of unshattered $d$-dimensional faces
for a randomly chosen subset $A\subseteq U^n$ of size
$\lceil d\log_b(n)+\log_b(u^d/d!)
\rceil$ and found it to be less than $1$, so at least one of the
sets of this size unshatters zero faces (i.e. shatters every face).
We don't have an explicit description of any
$\log(n)$-size fully-shattering sets.

[This combinatorial problem came up while investigating the
minimal sizes of generating
sets of powers $\mathbb U^n$ of a finite algebraic structure $\mathbb U$.]