Problem 20 — Visual explainer (undergrad)
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Problem statement
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Let $f(n,k)$ be minimal such that every family $\mathcal{F}$ of $n$-uniform sets with $\lvert \mathcal{F}\rvert \geq f(n,k)$ contains a $k$-sunflower. Is it true that\[f(n,k) < c_k^n\]for some constant $c_k>0$?

Picture
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      (petal)        (petal)
        \             /
         \           /
          \         /
           [  core  ]
          /         \
         /           \
        /             \
      (petal)        (petal)

Sunflower = many sets with the *same* intersection ("core"),
and the parts outside the core are disjoint ("petals").

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: extremal set theory / hypergraph combinatorics.

Benchmarks / known results (from comments)
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- Erd\H{o}s and Rado originally proved $f(n,k)\leq (k-1)^nn!$.
- Kostochka improved this slightly (in particular establishing an upper bound
  of $o(n!)$, for which Erd\H{o}s awarded him the consolation prize of \$100),
  but the bound stood at $n^{(1+o(1))n}$ for a long time until Alweiss,
  Lovett, Wu, and Zhang proved\[f(n,k) < (Ck\log n\log\log n)^n\]for some
  constant $C>1$.
- This was refined slightly, independently by Rao, Frankston, Kahn, Narayanan,
  and Park, and Bell, Chueluecha, and Warnke, leading to the current record
  of\[f(n,k) < (Ck\log n)^n\]for some constant $C>1$.
- In offered \$1000 for a proof or disproof even just in the special case when
  $k=3$, which he expected 'contains the whole difficulty'.