A Colouring Adversary

The Prompt

Write a program or carry out by hand an adversarial edge-colouring experiment for a small Ramsey problem. For R(3,3) = 6, demonstrate both sides: construct a two-colouring of K_5 that avoids monochromatic triangles, and prove that no such colouring exists for K_6. Then attempt the same for R(3,4) = 9 or R(4,4) = 18: find extremal colourings of the largest complete graph below the Ramsey number. Record your approach and what the adversarial process revealed about why larger Ramsey numbers are hard to compute.

Completion Criteria

The colourings produced (drawn, computed, or described), a proof or argument for the R(3,3) case, and a one-paragraph reflection on the difficulty encountered at larger values.

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