FormalConjectures.ErdosProblems.«624»

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import FormalConjecturesUtil

Erdős Problem 624

Reference: erdosproblems.com/624

namespace Erdos624


open Filter Finset

The condition that an integer m ensures the existence of a function f covering Fin n for all large enough subsets Y. The property is invariant under bijection, so we use a representative Fin n for a finite set of size n.

def ExistsEventuallySurjective (n m : ℕ) : Prop :=
  ∃ (f : Finset (Fin n) → Fin n),
    ∀ (Y : Finset (Fin n)), #Y ≥ m →
      Y.powerset.image f = Finset.univ

Let $H(n)$ be the minimum integer $m$ such that there is a function $f: \mathcal{P}(X) \to X$ where $X$ is a finite set of size $n$, such that for every subset $Y \subseteq X$ with $|Y| \ge m$, the set ${f(A) : A \subseteq Y}$ covers $X$.

noncomputable def H (n : ℕ) : ℕ :=
  if 0 < n then
    sInf {m : ℕ | ExistsEventuallySurjective n m}
  else 0

Let $X$ be a finite set of size $n$ and $H(n)$ be such that there is a function $f:{A : A\subseteq X}\to X$ so that for every $Y\subseteq X$ with $\lvert Y\rvert \geq H(n)$ we have $\left{ f(A) : A\subseteq Y\right}=X$. Prove that $H(n)-\log_2 n \to \infty$.

@[category research open, AMS 5]
theorem declaration uses 'sorry'erdos_624 :
    atTop.Tendsto (fun n : ℕ => H n - Real.logb 2 (n : ℝ)) atTop := by⊢ Tendsto (fun n => ↑(H n) - Real.logb 2 ↑n) atTop atTop
  sorryAll goals completed! 🐙


end Erdos624