Color every edge of the complete graph either red or blue. Ramsey theory guarantees that for large enough graphs, a monochromatic triangle is unavoidable. Can you avoid one?
Click an edge to color it. Choose red or blue above.
Ramsey's theorem: R(3,3)=6 means: color the edges of Kβ with 2 colors β there will always be a monochromatic triangle.
Proved by Frank Ramsey (1930). R(4,4)=18 (known), R(5,5)β[43,48] (still unknown after 90 years).
ErdΕs: "Suppose an alien demanded R(5,5) or they'd destroy Earth. We should gather all our computers and try to compute it.
If they asked for R(6,6), we should try to destroy the aliens."