$H(n+s)\le m+s$ implies that $\mathrm{TrackBad}(n,m,s)$ fails (Erdős 624)

Proved

Erdos624.not_trackBad_of_H_le

by Ethan Cheung Wang · Oct 23, 2026 · Mathlib 0df444a (Lean v4.33.1)

combinatoricserdos-problems

Let $H(n)$ be the least $m$ such that some $f:\mathcal P(X)\to X$ , $|X|=n$ , satisfies $\{f(A):A\subseteq Y\}=X$ for all $Y\subseteq X$ with $|Y|\ge m$ (Erdős Problem 624). Say $\mathrm{TrackBad}(n,m,s)$ holds if for every family of maps $g_B:\mathcal P(U)\to U$ ( $|U|=n$ ) indexed by the subsets $B$ of an $s$ -set, there are $Y\subseteq U$ with $|Y|=m$ and $u\in U$ with $u\notin\{g_B(A):A\subseteq Y\}$ for every $B$ . By an earlier result, $\mathrm{TrackBad}(n,m,s)$ implies $H(n+s)>m+s$ for $n\ge1$ . Theorem. If $n\ge1$ and $H(n+s)\le m+s$ then $\mathrm{TrackBad}(n,m,s)$ is false.

This is the contrapositive of track amplification: the track condition is never stronger than the target bound $H(n+s)>m+s$ itself.

Formalization Note. $U=\mathrm{Fin}\,n$ and the $s$ -set is $\mathrm{Fin}\,s$ , as in Erdos624.track_amplification.

Preamble

import Mathlib
import Definitions.Def_Erdos624_defs
open Finset

Formal statement

namespace Erdos624

theorem not_trackBad_of_H_le {n m s : ℕ} (hn : 0 < n) (h : H (n + s) ≤ m + s) :
    ¬ ∀ g : Finset (Fin s) → Finset (Fin n) → Fin n, ∃ Y : Finset (Fin n), #Y = m ∧
      ∃ u : Fin n, ∀ B : Finset (Fin s), ∀ A ∈ Y.powerset, g B A ≠ u := by sorry

end Erdos624

Source

Erdős Problem 624 (https://www.erdosproblems.com/624); definitions from Formal Conjectures 624.lean; Claim B2a of claim from GPT 6.1 Sol (second answer, Oct 23 2026), Lean-verified by Grok Bot for Ethan Cheung (Oct 23 2026)

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H(n+s)≤m+sH(n+s)\le m+sH(n+s)≤m+s implies that TrackBad(n,m,s)\mathrm{TrackBad}(n,m,s)TrackBad(n,m,s) fails (Erdős 624)

$H(n+s)\le m+s$ implies that $\mathrm{TrackBad}(n,m,s)$ fails (Erdős 624)