Problem 183 — Visual explainer (undergrad)
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Problem statement
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Let $R(3;k)$ be the minimal $n$ such that if the edges of $K_n$ are coloured with $k$ colours then there must exist a monochromatic triangle. Determine\[\lim_{k\to \infty}R(3;k)^{1/k}.\]

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 offers \$100 for showing that this limit is finite.
- An easy pigeonhole argument shows that\[R(3;k)\leq 2+k(R(3;k-1)-1),\]from
  which $R(3;k)\leq \lceil e k!\rceil$ immediately follows.
- The best-known upper bounds are all of the form $ck!+O(1)$, and arise from
  this type of inductive relationship and computational bounds for $R(3;k)$
  for small $k$.
- The best-known lower bound (coming from lower bounds for Schur numbers)
  is\[R(3,k)\geq (380)^{k/5}-O(1),\]due to Ageron, Casteras, Pellerin,
  Portella, Rimmel, and Tomasik (improving previous bounds of Exoo and
  Fredricksen and Sweet ).