Discrete Mathematics - Lecture notes https://www.isibang.ac.in/~d.yogesh/Course_Notes/DM1/main.pdf.8.4 Exercises 9.1 Examples and basic properties

Chapter 9 Designs

Design theory was motivated by statistical questions but now it is a subject of its own within combinatoricc and has a geometric flavour to it. We shall again see some basics of the theory.

Let $v,k,t,\lambda$ be integers such that $v\geq k\geq t\geq 0$ and $\lambda\geq 1$ . A $t-(v,k,\lambda)$ design consists of sets of $v$ points $\mathcal{P}$ (called as elements also), distinct $k$ -subsets ${\mathcal{B}}$ of $\mathcal{P}$ called as blocks satisfying the condition that for any set of $t$ points, there are exactly $\lambda$ blocks containing the $t$ points. In other words, $|\mathcal{P}|=v$ , $|B|=k$ for all $B\in{\mathcal{B}}$ . Also we denote $b=|{\mathcal{B}}|$ to be the number of blocks. Since $\mathcal{P}$ is often clear from context, we shall call ${\mathcal{B}}$ a $t-(v,k,\lambda)$ design. We drop $t$ if $t=2$ . Also observe that there is no condition to check for $t=0$ and in which case we take $b=\lambda$ .

A $t-(v,k,1)$ design is called Steiner system $S(t,k,v)$ . A design with $b=v$ is called a symmetric design .