Pure maths · Solid · Dissertation

From the pigeonhole principle to Ramsey numbers: a maths EPQ idea

Why it works as an EPQ

A one-line principle leads to surprising proofs and to a famous unsolved area, so you can map exactly where elementary methods stop.

Scope and difficulty

Solid. Solid. Prove R(3, 3) = 6 and a general bound; known values of larger Ramsey numbers belong in the discussion.

The maths

Builds on these A Level topics: Proof.

You would learn:

• Graph colourings
• Ramsey numbers
• Bounds by induction

Learning maths beyond the course is expected in a maths EPQ, but you must understand and explain everything you use.

One possible plan

1. Collect and prove pigeonhole results of increasing difficulty.
2. Prove that any 6 people include 3 mutual friends or 3 mutual strangers, and that 5 are not enough.
3. Prove an upper bound for R(m, n) by induction.
4. Explain why exact values are so hard to find.

Pitfalls

• A list of puzzles with no thread.
• Claiming values of Ramsey numbers you have not checked in a reliable source.

Where to start reading

• Search for: Ramsey number R(3,3)=6 proof
• Search for: pigeonhole principle olympiad problems

Prefer books, lecture notes, journal articles and official data to a single website, and record every source as you read (how to reference).

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