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

TL;DR

This paper introduces a polynomial ideal associated with $t$-$(v,k,\lambda)$ designs, analyzing its algebraic properties and parameters to distinguish different combinatorial design structures.

Contribution

It defines and investigates the parameters $oldsymbol{eta_1}$ and $oldsymbol{eta_2}$ of the ideal related to $t$-designs, establishing bounds and exploring their behavior across various design families.

Findings

Bounds established: $t/2 < eta_1 \,\le\, \beta_2 \le k$

Symmetric 2-designs and Steiner systems have $eta_2 \le t$

Constructed infinite triple systems with $eta_2 = k$

Abstract

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 $\gamma_2(\mathcal{B})=k$.