Color edges of K_n red/blue — for n≥6 you always get a monochromatic triangle (R(3,3)=6)
Click an edge to toggle red/blue. Find a valid 2-coloring for n < 6!
R(3,3) = 6
R(3,4) = 9
R(3,5) = 14
R(4,4) = 18
R(5,5) = 43–48
Ramsey's Theorem: For any graph coloring, large enough complete graphs must contain monochromatic cliques. R(3,3)=6 means: any 2-coloring of K_6 contains a monochromatic triangle — provably unavoidable. For K_5 it IS possible to avoid. The upper bound R(s,t) ≤ C(s+t-2, s-1) comes from a probabilistic argument; exact values are notoriously hard to compute. R(5,5) is still unknown after 70+ years.