Editing Ramsey's theorem (section)

=== A multicolour example: ''R''(3, 3, 3) = 17 ===
{{multiple image
|image1=K 16 partitioned into three Clebsch graphs.svg
|image2=K 16 partitioned into three Clebsch graphs twisted.svg
|footer=The only two 3-colourings of {{math|''K''{{sub|16}}}} with no monochromatic {{math|''K''{{sub|3}}}}, up to isomorphism and permutation of colors: the untwisted (left) and twisted (right) colorings.
}}

A multicolour Ramsey number is a Ramsey number using 3 or more colours. There are (up to symmetries) only two non-trivial multicolour Ramsey numbers for which the exact value is known, namely {{math|1=''R''(3, 3, 3) = 17}} and {{math|1=''R''(3, 3, 4) = 30}}.<ref name="bananas" />

Suppose that we have an edge colouring of a complete graph using 3 colours, red, green and blue. Suppose further that the edge colouring has no monochromatic triangles. Select a vertex {{mvar|v}}. Consider the set of vertices that have a red edge to the vertex {{mvar|v}}. This is called the red neighbourhood of {{mvar|v}}. The red neighbourhood of {{mvar|v}} cannot contain any red edges, since otherwise there would be a red triangle consisting of the two endpoints of that red edge and the vertex {{mvar|v}}. Thus, the induced edge colouring on the red neighbourhood of {{mvar|v}} has edges coloured with only two colours, namely green and blue. Since {{math|1=''R''(3, 3) = 6}}, the red neighbourhood of {{mvar|v}} can contain at most 5 vertices. Similarly, the green and blue neighbourhoods of {{mvar|v}} can contain at most 5 vertices each. Since every vertex, except for {{mvar|v}} itself, is in one of the red, green or blue neighbourhoods of {{mvar|v}}, the entire complete graph can have at most {{nowrap|1=1 + 5 + 5 + 5 = 16}} vertices. Thus, we have {{math|''R''(3, 3, 3) ≤ 17}}.

To see that {{math|1=''R''(3, 3, 3) = 17}}, it suffices to draw an edge colouring on the complete graph on 16 vertices with 3 colours that avoids monochromatic triangles. It turns out that there are exactly two such colourings on {{math|''K''{{sub|16}}}}, the so-called untwisted and twisted colourings. Both colourings are shown in the figures to the right, with the untwisted colouring on the left, and the twisted colouring on the right.

[[File:Clebsch graph.svg|thumb|right|[[Clebsch graph]]]]

If we select any colour of either the untwisted or twisted colouring on {{math|''K''{{sub|16}}}}, and consider the graph whose edges are precisely those edges that have the specified colour, we will get the [[Clebsch graph]].

It is known that there are exactly two edge colourings with 3 colours on {{math|''K''{{sub|15}}}} that avoid monochromatic triangles, which can be constructed by deleting any vertex from the untwisted and twisted colourings on {{math|''K''{{sub|16}}}}, respectively.

It is also known that there are exactly 115 edge colourings with 3 colours on {{math|''K''{{sub|14}}}} that avoid monochromatic triangles, provided that we consider edge colourings that differ by a permutation of the colours as being the same.