# A new relationship between block designs

## TL;DR

This paper introduces a novel method for constructing new block designs through intersection-based procedures and explores a symmetric friendship relationship among designs, leading to the concept of friendly families and their partial ordering.

## Contribution

It presents a new intersection-based construction method for block designs and defines the concept of friendly families with a partial order, expanding the theoretical framework of block design relationships.

## Key findings

- Introduction of a symmetric friendship relationship between block designs
- Existence of a partial order within friendly families
- A mapping from the power set of V to a friendly family that preserves partial order

## Abstract

We propose a procedure of constructing new block designs starting from a given one by looking at the intersections of its blocks with various sets and grouping those sets according to the structure of the intersections. We introduce a symmetric relationship of friendship between block designs built on a set $V$ and consider families of block designs where all designs are friends of each other, the so-called friendly families. We show that a friendly family admits a partial ordering. Furthermore, we exhibit a map from the power set of $V$, partially ordered by inclusion, to a friendly family of a particular type which preserves the partial order.