Problem 593 — Visual explainer (undergrad)
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Problem statement
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Characterize those finite 3-uniform hypergraphs which appear in every 3-uniform hypergraph of chromatic number $>\aleph_0$.

Picture
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Graph picture:

o---o
|\  |
| \ |
o---o    K4 is the complete graph on 4 vertices (all connections)

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).

Context
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Area hint: infinite graph theory / Ramsey-type decompositions.

Benchmarks / known results (from comments)
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- Erd\H{o}s claims that for graphs the problem is completely solved: a graph
  of chromatic number $\geq \aleph_1$ must contain all finite bipartite graphs
  but need not contain any fixed odd cycle.