stylistic:tone
Hypergraphs

The theorem can also be extended to hypergraphs. An m -hypergraph is a graph whose "edges" are sets of m vertices – in a normal graph an edge is a set of 2 vertices. The full statement of Ramsey's theorem for hypergraphs is that for any integers m and c, and any integers n, …, n, there is an integer R (n, …, n; m) such that if the hyperedges of a complete m -hypergraph of order R (n, …, n; m) are coloured with c different colours, then for some i between 1 and c, the hypergraph must contain a complete sub- m -hypergraph of order n whose hyperedges are all colour i. This theorem is usually proved by induction on m, the 'hyper-ness' of the graph. The base case for the proof is m = 2, which is exactly the theorem above.

For m = 3 we know the exact value of one non-trivial Ramsey number, namely R (4, 4; 3) = 13. This fact was established by Brendan McKay and Stanisław Radziszowski in 1991. Additionally, we have: R (4, 5; 3) ≥ 35, R (4, 6; 3) ≥ 63 and R (5, 5; 3) ≥ 88.