Problem 77 — Visual explainer (undergrad)
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Problem statement
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If $R(k)$ is the Ramsey number for $K_k$, the minimal $n$ such that every $2$-colouring of the edges of $K_n$ contains a monochromatic copy of $K_k$, then find the value of\[\lim_{k\to \infty}R(k)^{1/k}.\]

Picture
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[Given objects + constraints]  --->  [Count / structure]  --->  [Show bound / existence]

Conjecture / Task: what to look at
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- Restate the problem as: given the constraint, what asymptotic bound or
  structure must follow?
- Use the comments as benchmarks (best known bounds / constructions).

Benchmarks / known results (from comments)
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- Erd\H{o}s offered \$100 for just a proof of the existence of this constant,
  without determining its value.
- He also offered \$1000 for a proof that the limit does not exist, but says
  'this is really a joke as [it] certainly exists'.
- (In he raises this prize to \$10000).
- Erd\H{o}s proved\[\sqrt{2}\leq \liminf_{k\to \infty}R(k)^{1/k}\leq
  \limsup_{k\to \infty}R(k)^{1/k}\leq 4.\]The upper bound has been improved to
  $4-\tfrac{1}{128}$ by Campos, Griffiths, Morris, and Sahasrabudhe.