Problem 592 — Visual explainer (undergrad)
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Problem statement
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Determine which countable ordinals $\beta$ have the property that, if $\alpha=\omega^{^\beta}$, then in any red/blue colouring of the edges of $K_\alpha$ there is either a red $K_\alpha$ or a blue $K_3$.

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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- Such $\alpha$ are called partition ordinals.
- {UL} {LI}Specker proved this holds for $\beta=2$ and not for $3\leq \beta
  <\omega$.{/LI} {LI} Chang proved this holds for $\beta=\omega$.{/LI} {LI}
  Galvin and Larson have shown that if $\beta\geq 3$ has this property then
  $\beta$ must be 'additively indecomposable', so that in particular
  $\beta=\omega^\gamma$ for some countable ordinal $\gamma$.
- Galvin and Larson conjecture that every $\beta\geq 3$ of this form has this
  property.{/LI} {LI}Schipperus have proved this is holds if
  $\beta=\omega^\gamma$ in which $\gamma$ is a countable ordinal which is the
  sum of one or two indecomposable ordinals, and this fails to hold if
  $\gamma$ is the sum of four or more indecomposable ordinals.{/LI} {/UL} The
  remaining open case appears to be when $\gamma$ is the sum of three
  indecomposable ordinals.
- The case $\beta=\omega$ is the subject of [590], and $\beta=\omega^2$ is the
  subject of [591].