# Incidence system

This term is related to: incidence geometry
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## Definition

An incidence system $(G,*,t,I)$ is the following data:

• A set of elements $G$
• A reflexive symmetric binary relation $*$ called the incidence relation
• A set $I$ of possible types
• A type function $t:G \to I$ which essentially partitions the elements of $G$ into types, such that no two elements of the same type are incident on each other

In the language of graph theory, an incidence system can be thought of as a $|I|$-partite graph structure with vertex set $G$ and the various parts being $t^{-1}(i)$ for $i \in I$.

## Terminology

### Rank of an incidence system

The rank of an incidence system is defined as the cardinality of the image of $t$. This roughly corresponds to the notion of chromatic number in graph theory.

### Flag

A flag is a set of pairwise incident elements. This corresponds to a clique in the corresponding graph.

The type of a flag is the set of types of all elements in it, and the rank of a flag is the number of elements in it.

The cotype is the complement of the type in the set $I$ and the corank is the size of the cotype.