# New lower bounds for $t$-coverings

## TL;DR

This paper introduces new lower bounds for the size of t-coverings in combinatorial design theory, extending previous bounds and employing combined methods from Bose and Wilson.

## Contribution

It presents novel lower bounds for t-coverings that generalize recent results on 2-coverings, using combined proof techniques from earlier foundational work.

## Key findings

- New lower bounds for t-coverings established
- Generalization of bounds for 2-coverings
- Improved understanding of covering sizes in combinatorial designs

## Abstract

Fisher proved in 1940 that any $2$-$(v,k,\lambda)$ design with $v>k$ has at least $v$ blocks. In 1975 Ray-Chaudhuri and Wilson generalised this result by showing that every $t$-$(v,k,\lambda)$ design with $v \geq k+\lfloor t/2 \rfloor$ has at least $\binom{v}{\lfloor t/2 \rfloor}$ blocks. By combining methods used by Bose and Wilson in proofs of these results, we obtain new lower bounds on the size of $t$-$(v,k,\lambda)$ coverings. Our results generalise lower bounds on the size of $2$-$(v,k,\lambda)$ coverings recently obtained by the first author.