Problem 132 — Visual explainer (undergrad)
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Problem statement
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Let $A\subset \mathbb{R}^2$ be a set of $n$ points. Must there be two distances which occur at least once but between at most $n$ pairs of points? Must the number of such distances $\to \infty$ as $n\to \infty$?

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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- Hopf and Pannowitz proved that the largest distance between points of $A$
  can occur at most $n$ times, but it is unknown whether a second such
  distance must occur.
- It may be true that there are at least $n^{1-o(1)}$ many such distances.
- In Erd\H{o}s offers \$100 for 'any nontrivial result'.
- Erd\H{o}s believed that for $n\geq 5$ there must always exist at least two
  such distances.