Erdős problem 107 · problems.science

Let $f(n)$ be minimal such that any $f(n)$ points in $ℝ^2$ , no three on a line, contain $n$ points which form the vertices of a convex $n$ -gon. Prove that $f(n) = 2^{n-2} + 1$ .

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7 retained statements2415f78e850a

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FormalConjectures/ErdosProblems/

107.lean

Retained formal statement6 of 7

The current best bound is due to Holmsen, Mojarrad, Pach, and Tardos [HMPT20], who prove $f(n) ≤ 2^{n+O(\sqrt{n\log n})}.$

[HMPT20] Holmsen, Andreas F. and Mojarrad, Hossein Nassajian and Pach, János and Tardos, Gábor, _Two extensions of the Erdős-Szekeres problem_. J. Eur. Math. Soc. (JEMS) (2020), 3981-3995.

FormalConjectures/ErdosProblems/107.leanErdos107.variants.hmpt_bound1 lineExact file

1∃ r, (r =O[Filter.atTop] fun n => √(↑n * Real.log ↑n)) ∧ ∀ n ≥ 3, ↑(Erdos107.f n) ≤ 2 ^ (↑n + r n)

SolvedStatement only, no proof

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