An incidence geometry $(P,\mathcal{B})$ consists of a set of $v$ points $P=\{{p}_{1},\cdots ,{p}_{v}\}$ and a collection of $b$ blocks $\mathcal{B}=\{{B}_{1},\cdots ,{B}_{b}\}$ such that ${B}_{i}\subseteq P$ for $i=1,2,\cdots ,b$. The number of blocks containing a point $p\in P$ is called the degree, denoted by $\left[p\right]$. Blocks are also called lines and $\left|B\right|$ is called the length of the block $B$. A pair $(p,B)$ with $p\in B\in \mathcal{B}$ is called a flag. In this case, the point $p$ is said to lie on the line $B$. The line is said to pass through $p$. Additionally, in an incidence geometry any pair of points is joined by at most one line, i.e., $|{B}_{i}\cap {B}_{j}|\le 1$ for all $i,j\in \{1,\cdots ,b\}$, $i\ne j$.

An incidence geometry $(p,\mathcal{B})$ is called configuration of type $({v}_{r},{b}_{k})$ if (1) $|{B}_{j}|=k$ for $j=1,2,\cdots ,b$ and (2) $\left[{p}_{i}\right]=r$ for $i=1,2,\cdots ,v$. A configuration $(p,\mathcal{B})$ with $v=b$ is called symmetric. Its type is simply denoted by ${v}_{r}$ which is the same as ${b}_{k}$ because $k=r$ in this case. A configuration $C=(p,\mathcal{B})$ is called decomposable if it can be written as the union of two configurations ${C}_{1}$, ${C}_{2}$ on distinct point sets: ${C}_{1}=({p}_{1},{\mathcal{B}}_{1})$, ${C}_{2}=({p}_{2},{\mathcal{B}}_{2})$, and $P={P}_{1}\cup {P}_{2}$ and $\mathcal{B}={\mathcal{B}}_{1}\cup {\mathcal{B}}_{2}$. Indecomposable configurations are also called connected. The isomorphisms are structure preserving mappings.

A triangle of a configuration consists of three points, say $a$, $b$, and $c$, such that the three pairs $\{a,b\}$, $\{b,c\}$, $\{c,a\}$ are contained in different blocks. A configuration that has no triangles is called a triangle-free configuration. An isomorphism $C\to {C}^{d}$ is called an anti-automorphism. A configuration which admits an anti-automorphism is called self-dual. An anti-automorphism of order 2 is called a polarity. A configuration which admits a polarity is called self-polar.

The authors give tables of configurations ${v}_{3}$ for $v\le 18$ and triangle-free configurations for $v\le 21$ together with some statistics about some properties of structures like transitivity, self-duality or self-polarity.

##### MSC:

05B30 | Other designs, configurations |