Infinite designs and Fisher's Inequality
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
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.