# Wilson's Theorem

Theorem [1]

For any given graph $G$, there exists an integer $N(G)$ ($N(G)$ is exponentially large) such that if

• $n > N(G)$;
• $n(n-1) \equiv 0 \,({\rm mod \ }2|E(G)|)$; and
• $n-1 \equiv 0 \,({\rm mod \ }d)$ where $d$ is the greatest common divisor of the degrees of the vertices in $G$,

then there exists a $G$-design of order $n$.

1. Wilson, R. M. Decompositions of complete graphs into subgraphs isomorphic to a given graph, Proceedings of the Fifth British Combinatorial Conference (Univ. Aberdeen, Aberdeen, 1975), Congressus Numerantium, No. XV, Utilitas Math., Winnipeg, Man. 647–659 (1976). Bib