With the random-graph definition of the Rado graph, each vertex outside the union of "U" and "V" has probability 1/2|"U"| + |"V"| of fulfilling the extension property, independently of the other vertices. Because there are infinitely many vertices to choose from, each with the same finite probability of success, the probability is one that there exists a vertex that fulfils the extension property. With the Paley graph definition, for any sets "U" and "V", by the Chinese remainder theorem, the numbers that are quadratic residues modulo every prime in "U" and nonresidues modulo every prime in "V" form a periodic sequence, so by Dirichlet's theorem on primes in arithmetic progressions this number-theoretic graph has the extension property.