Problem
COM-B2-M06-P006 A Monochromatic Triangle in \(K_6\)
#6
★★★☆☆ Level 3 of 5
Prove that every red-blue colouring of the edges of the complete graph \(K_6\) contains a monochromatic triangle.
Choose a vertex and find three incident edges of one colour.
Choose a vertex \(v\). From it, \(5\) edges leave, so at least \(3\) have the same colour. Let these be red edges to vertices \(a,b,c\).
If at least one of the edges \(ab,bc,ca\) is red, it forms a red triangle with \(v\). If none of these edges is red, then all of them are blue, and vertices \(a,b,c\) form a blue triangle.
The main classical problem of the module.