Problem
COM-B2-M06-P003 A Monochromatic Path of Length Two
#3
★★☆☆☆ Level 2 of 5
Prove that in every red-blue colouring of the edges of \(K_4\), there is a path consisting of two edges of the same colour.
Look at the three edges leaving one vertex.
Choose a vertex \(v\). From it, \(3\) edges leave in two colours. By the pigeonhole principle, two of them have the same colour, say \(va\) and \(vb\).
Then \(a-v-b\) is a path of two edges of one colour.
Shows the difference between a weak and a strong forced structure.