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import FormalConjecturesUtilErdős Problem 624
Reference: erdosproblems.com/624
namespace Erdos624open 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.univLet $H(n)$ 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 0Let $X$ 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 erdos_624 :
atTop.Tendsto (fun n : ℕ => H n - Real.logb 2 (n : ℝ)) atTop := ⊢ Tendsto (fun n ↦ ↑(H n) - Real.logb 2 ↑n) atTop atTop
All goals completed! 🐙end Erdos624