Infinite graphs

In graph theory, an **infinite graph** is a graph that has an infinite number of vertices, edges, or both. Unlike finite graphs, which have a limited number of vertices and edges, infinite graphs can be more complex and often require different techniques for analysis and study. ### Key Characteristics of Infinite Graphs: 1. **Infinite Vertices**: An infinite graph can have an infinite number of vertices.

The Henson graph is an important concept in the field of graph theory, particularly in the study of countable structures and model theory. It is named after the mathematician John Henson who introduced it in the context of descriptive set theory and the study of universal structures. The Henson graph can be defined as follows: - It is a **countable graph** that is **triangle-free**, meaning that it does not contain any triangles (three vertices that are all mutually connected).

Kőnig's lemma is a result in set theory and combinatorics, particularly in the context of infinite trees. It states that: If every infinite, finitely branching tree has an infinite path, then the tree must have an infinite path. More formally, Kőnig's lemma can be stated as follows: Let \( T \) be a tree such that: 1. Every node in \( T \) has finitely many children (i.e.