A polynomial ideal associated to any $t$-$(v,k,\lambda )$ design

We consider ordered pairs $(X,\mathcal{B})$ where $X$ is a finite set of size $v$ and $\mathcal{B}$ is some collection of $k$-element subsets of $X$ such that every $t$-element subset of $X$ is contained in exactly $\lambda$ “blocks” $B\in \mathcal{B}$ for some fixed $\lambda$. We represent each block $B$ by a zero-one vector ${\mathbf{c}}_B$ of length $v$ and explore the ideal $\mathcal{I}(\mathcal{B})$ of polynomials in $v$ variables with complex coefficients which vanish on the set ${ $\mathbf{c}$_B \mid B \in $\mathcal{B}$}$. After setting up the basic theory, we investigate two parameters related to this ideal: ${\gamma }_1(\mathcal{B})$ is the smallest degree of a non-trivial polynomial in the ideal $\mathcal{I}(\mathcal{B})$ and ${\gamma }_2(\mathcal{B})$ is the smallest integer $s$ such that $\mathcal{I}(\mathcal{B})$ is generated by a set of polynomials of degree at most $s$. We first prove the general bounds $t/2 < \gamma_1($\mathcal{B}$) \le \gamma_2($\mathcal{B}$) \le k$. Examining important families of examples, we find that, for symmetric 2-designs and Steiner systems, we have ${\gamma }_2(\mathcal{B})\le t$. But we expect ${\gamma }_2(\mathcal{B})$ to be closer to $k$ for less structured designs and we indicate this by constructing infinitely many triple systems satisfying ${\gamma }_2(\mathcal{B})=k$.