 Rado graph

The Rado graph is the unique countably infinite graph with the two-uniform hypergraph extension property. Equivalently, for any disjoint finite vertex sets $U,W$, some new vertex is adjacent to every member of $U$ and no member of $W$. The back-and-forth method proves uniqueness. Independent edge sampling on the natural numbers with any fixed $p\in (0,1)$ produces this graph almost surely. It contains both edges and nonedges and therefore is not a complete graph. Its first-order axioms constitute the theory of the random graph.

The Rado graph, also known as the Random graph or Rado's graph, is a specific type of infinite, countably infinite graph that is unique up to isomorphism. It is named after the mathematician Richard Rado. Here are some key attributes and characteristics of the Rado graph: 1. **Countably Infinite**: The graph has a countably infinite number of vertices. 2. **Universal Graph**: The Rado graph is universal for all countable graphs.