Problem 99 — Visual explainer (undergrad)
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Problem statement
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Let $A\subseteq\mathbb{R}^2$ be a set of $n$ points with minimum distance equal to 1, chosen to minimise the diameter of $A$. If $n$ is sufficiently large then must there be three points in $A$ which form an equilateral triangle of size 1?

Picture
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Plane picture:

y ^
  |    o      o
  |        o
  | o
  +-----------------> x

Question: 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: incidence geometry / combinatorial geometry.

Benchmarks / known results (from comments)
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- Thue proved that the minimal such diameter is achieved (asymptotically) by
  the points in a triangular lattice intersected with a circle.
- In general Erd\H{o}s believed such a set must have very large intersection
  with the triangular lattice (perhaps as many as $(1-o(1))n$).
- Simonovits doubted the truth of this conjecture.' In he offers \$100 for a
  counterexample but only \$50 for a proof.
- The stated problem is false for $n=4$, for example taking the points to be
  vertices of a square.