Editing Ramsey's theorem (section)

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

For {{math|1=''m'' = 3}} we know the exact value of one non-trivial Ramsey number, namely {{math|1=''R''(4, 4; 3) = 13}}. This fact was established by Brendan McKay and Stanisław Radziszowski in 1991.<ref>{{Cite journal |last1=McKay |first1=Brendan D. |last2=Radziszowski |first2=Stanislaw P. |date=1991 |title=The First Classical Ramsey Number for Hypergraphs is Computed |journal=Proceedings of the Second Annual ACM-SIAM Symposium on Discrete Algorithms, SODA'91 |pages=304–308}}</ref> Additionally, we have: {{math|''R''(4, 5; 3) ≥ 35}},<ref name=Dybizbański2018>{{Cite journal
| last=Dybizbański | first=Janusz
| date=2018-12-31
| title=A lower bound on the hypergraph Ramsey number R(4,5;3)
| journal=Contributions to Discrete Mathematics
| language=en
| volume=13
| issue=2
| doi=10.11575/cdm.v13i2.62416 | doi-access=free
| issn=1715-0868}}</ref> {{math|''R''(4, 6; 3) ≥ 63}} and {{math|''R''(5, 5; 3) ≥ 88}}.<ref name=Dybizbański2018 />