Understanding the Steiner triple system

Question

I am learning what Steiner systems are and I stumboled upon this Wolfram Mathworld source: https://mathworld.wolfram.com/SteinerTripleSystem.html.

I understand that $k = 3$ means that there are exactly 3 points(elements of set $X$) on each block. I also understand that the first parameter means how many points there are altogether. However I do not understand what is $\lambda$ that is the third parameter of $S$ in $S (v)=S(v,k=3,\lambda = 1)$

Answer 1

These are some of the parameters of a block design. Quoting Wikipedia:

• $v$ is the number of points (elements of $X$).
• $b$, not appearing in the article you're quoting, is the number of blocks: the subsets of $X$ we're taking.
• $r$, also not appearing in the article, is the number of blocks containing a given point.
• $k$ is the number of points in a block (for a Steiner triple system, $k=3$).
• $\lambda$ is the number of blocks containing...
• ...any $t$ distinct points. In the case of a Steiner triple system, $\lambda=1$ and $t=2$: for any two distinct points, there is exactly one subset containing them.