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Gallai’s colouring theorem states that if the edges of a complete graph are 3-coloured, with each colour class forming a connected (spanning) subgraph, then there is a triangle that has all three colours. What happens for more colours: if we $$k$$ k -colour the edges of the complete graph, with each colour class connected, how many of the $$\left( {\begin{array}{c}k\\ 3\end{array}}\right)...
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