Ramsey number R(s,t): minimum n so any 2-coloring of Kₙ edges contains a red Kₛ or blue Kₜ. R(3,3)=6 (party problem: among 6 people, 3 mutual friends or 3 mutual strangers exist). Very few exact values known: R(4,4)=18, R(4,5)=25. R(5,5) is unknown!
How to Use
Enter s and t:
- R(s,t): Exact value if known
- Bounds: Lower and upper
- Table: Known values matrix
Party Problem
R(3,3)=6: at a party of 6, there must be 3 mutual friends or 3 mutual strangers. With 5 people, a counterexample exists (C₅ coloring). This is the simplest Ramsey result.
Why So Hard?
Erdős: 'Imagine aliens demand R(5,5) or they destroy Earth. We should marshal our computer resources. If they ask for R(6,6), we should prepare for war.' Even R(5,5) has bounds 43≤R(5,5)≤48. The gap shrinks slowly.
Step-by-Step Instructions
- 1Enter s and t.
- 2Look up R(s,t).
- 3Check bounds.
- 4View Ramsey table.
- 5Explore party problem.