For a family $\f$
of sets and a real number $l\geq 0$, let $\ddeg{\f}{l}$ denote the
maximal possible number of sets in $\f$ containing a fixed set with
$l$ (or more) elements. In other words, $\ddeg{\f}{l}$ is the maximal
possible number of sets in $\f$ whose intersection has $l$ (or more)
elements. In particular, if $m$ is the maximum cardinality of a set
of $\f$, then $|\f|=\ddeg{\f}{0}\geq \ddeg{\f}{1}\geq \ldots\geq
\ddeg{\f}{m}=1$, and $\ddeg{\f}{l}=0$ for all $l>m$. Note that a
nonempty $m$-uniform family $\f$ is an $(m,d)$-design if and only if
$\ddeg{\f}{d}=1$. Also, $\ddeg{\f}{1}=1$ means that all sets of $\f$
are disjoint.
Let $m$ be a prime power, $1\leq d\leq m$ an
integer, and consider the grid $\gf{m}\times\gf{m}$. The
polynomial $(m,d)$-design $\f$ consists of all $|\f|=m^d$
subsets $S$ of points in this grid of the form $S=\{(a,p(a))\colon
a\in \gf{m}\}$ for a univariate polynomial $p=p(x)$ of degree at most
$d-1$ over $\gf{m}$. Note that no two points of any of these sets $S$
lie in the same row of the grid.
The main combinatorial property of polynomial designs is the
following.
Lemma:
Let $\f$ be a polynomial $(m,d)$-design, and $1\leq d\leq m$. For
every set of $l\leq d$ points of the grid $\gf{m}\times\gf{m}$, with
no two in the same row, exactly $m^{d-l}$ sets of $\f$ contain this
set. In particular, $\ddeg{\f}{l}\leq m^{d-l}$ holds for every
$0\leq l\leq d$.
Proof:
This is a direct consequence of a standard result in polynomial
interpolation. For any $l\leq d$ distinct points
$(a_1,b_1),\ldots,(a_l,b_l)$ in $\gf{m}\!\times\! \gf{m}$, the
number of polynomials $p(x)$ of degree at most $d-1$ satisfying
$p(a_1)=b_1,\ldots,p(a_l)=b_l$ is either $0$ (if $a_i=a_j$ holds for
some $i\neq j$) or is exactly $m^{d-l}$: this latter number is exactly the number of solutions of the
corresponding system of linear equations, with coefficients of $p$
viewed as variables.
$\Box$