Problem 101 — Visual explainer (undergrad)
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Problem statement
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Given $n$ points in $\mathbb{R}^2$, no five of which are on a line, the number of lines containing four points is $o(n^2)$.

Picture
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Many points on a line:

o---o---o---o---o
\
 o   o      o

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: incidence geometry / combinatorial geometry.

Notation (if it appears above)
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- `o(g(n))` means "grows strictly smaller than g(n)" (ratio → 0 as n→∞).

Benchmarks / known results (from comments)
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- There are examples of sets of $n$ points with $\sim n^2/6$ many collinear
  triples and no four points on a line.
- Such constructions are given by Burr, Gr\"{u}nbaum, and Sloane and
  F\"{u}redi and Pal\'{a}sti.
- Gr\"{u}nbaum constructed an example with $\gg n^{3/2}$ such lines.
- This is false: Solymosi and Stojakovi\'{c} have constructed a set with no
  five on a line and at least\[n^{2-O(1/\sqrt{\log n})}\]many lines containing
  exactly four points.