Adding 2 to $P^*$ needs at least three auxiliary letters

Proved

Erdos195.pstar_two_three_aux

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

arithmetic-progressionscombinatoricserdos-problems

Tree orders: for a tree witness $\theta(k,r)\in\{0,1\}$ and distinct integers $x,y$ with $k=\nu_2(y-x)$ , $x\prec_\theta y$ iff $\mathrm{bit}_k(x)=\theta(k,\,x\bmod 2^k)$ . A finite word $Q$ is $\theta$ -safe if " $Q$ in order, then all other integers in $\prec_\theta$ -order" has no monotone $4$ -term arithmetic progression; it is tree-safe if it is $\theta$ -safe for some $\theta$ . An in-order pair $a$ before $b$ in $Q$ is open if $2b-a,3b-2a\notin Q$ . Let $P^*=(0,1,-1,-5)$ . If $Q$ is $\theta$ -safe, begins with $P^*$ and contains $2$ , then $3$ precedes $2$ , and either $Q$ meets both $\{5,8\}$ and $\{9,16\}$ , or $Q$ meets both $\{-2,-3\}$ and $\{-10,-15\}$ . Consequently $Q$ has at least three distinct letters outside $\{-5,-1,0,1,2\}$ .

Formalization note: treeLt is from Definitions.Def_Erdos195_tree; Safe, bitOne from Definitions.Def_Erdos195_defs; positions are List.idxOf, $\nu_2$ is padicValInt 2, L <+: Q means $L$ is a prefix of $Q$ .

Preamble

import Mathlib
import Definitions.Def_Erdos195_tree

Formal statement

namespace Erdos195

theorem pstar_two_three_aux (θ : ℕ → ℤ → Bool) (Q : List ℤ) (hS : Safe (treeLt θ) Q)
    (hpre : [0, 1, -1, -5] <+: Q) (h2 : (2 : ℤ) ∈ Q) :
    (3 ∈ Q ∧ Q.idxOf 3 < Q.idxOf 2) ∧
    (((5 ∈ Q ∨ 8 ∈ Q) ∧ (9 ∈ Q ∨ 16 ∈ Q)) ∨ ((-2 ∈ Q ∨ -3 ∈ Q) ∧ (-10 ∈ Q ∨ -15 ∈ Q))) ∧
    ∃ x ∈ Q, ∃ y ∈ Q, ∃ z ∈ Q, x ≠ y ∧ x ≠ z ∧ y ≠ z ∧
      x ∉ [-5, -1, 0, 1, 2] ∧ y ∉ [-5, -1, 0, 1, 2] ∧ z ∉ [-5, -1, 0, 1, 2] := by sorry

end Erdos195

Source

Erdős Problem 195 (https://www.erdosproblems.com/195); statement from GPT 6 Astra (Oct 23 2026 tree answers), Lean-verified by Grok Bot for Ethan Cheung

View graph

Adding 2 to P∗P^*P∗ needs at least three auxiliary letters

Adding 2 to $P^*$ needs at least three auxiliary letters