Infinite designs and Fisher's Inequality

Bridget Webb
Open University

A $t$-$(v,k,\lambda)$ design comprises a $v$-set of points with a collection of $k$-subsets called blocks, with every $t$-set of points contained in precisely $\lambda$ blocks.

We look at infinite designs---designs with infinitely many points. There can be non-trivial designs with $v=k$, so in addition we require that no block contains every point. We show that this definition does not rule out dull structures. Camina suggested adding regularity conditions to the definition and we propose strengthening these conditions and adding a uniformity condition.

We show that all $t$-$(v,k,\lambda)$ designs with $t$ and $\lambda$ finite satisfy this definition. Thus we are tightening the definition only to rule out anomolous structures where $t$ and $\lambda$ are not both finite.

A fundamental result for finite $2$-designs is Fisher's Inequality, which states that, for every finite $2$-design, $b \geq v$. We give an infinite analogue to this result.