Problem
COM-B2-M06-P011 The Value of \(R(2,t)\)
#11
★★★☆☆ Level 3 of 5
Prove that \(R(2,t)=t\) for every \(t\ge2\).
A red \(K_2\) is just a red edge.
First show that \(t\) vertices are enough. In a red-blue colouring of \(K_t\), if there is a red edge, then there is a red \(K_2\). If there are no red edges, then all edges are blue, and all \(t\) vertices form a blue \(K_t\).
Fewer than \(t\) vertices are not enough: colour all edges blue. Then there is no red \(K_2\), and there is no blue \(K_t\) because there are fewer than \(t\) vertices. Hence \(R(2,t)=t\).
A necessary small base for Ramsey recursion.