Restricted reciprocal Rado numbers

Abstract

For integers k ≥ 2, r ≥ 2 and 1 ≤ ℓ ≤ k, let fᵣ(k;ℓ) denote the least n such that every r-colouring of {1,…,n} admits a monochromatic solution of (1)/(x₁)+(1)/(x₂)+…+(1)/(xₖ)=frac(1)xₖ₊₁ in which the number of distinct integers among x₁,…,xₖ₊₁ is exactly ℓ+1. Dropping the distinctness condition gives the reciprocal Rado number fᵣ(k), finite by Brown–Rödl and Lefmann and computed in small cases by Myers–Parrish and by Gaiser–Ramezanpour; the level ℓ was introduced by Gaiser for the linear equation x₁+…+xₖ=xₖ₊₁. The two have not been combined, and the combination is posed here for the first time. We prove that fᵣ(k;1) exists for no k ≥ 2 and no r ≥ 2, by the explicit two-colouring v ↦ ⌊logₖ v⌋ mod 2; the point is that the level-1 solutions of the reciprocal equation are exactly the pairs {a,ka}, which is literally the same set as for the linear equation. We determine the six values f₂(2;2)=120, f₂(3;2)=84, f₂(3;3)=126, f₂(4;2)=108, f₂(4;3)=90 and f₂(4;4)=180, each by an avoiding colouring at n-1 checked against a provably complete solution list together with a refutation at n. In particular f₂(k;ℓ) is not monotone in ℓ — f₂(4;2)=108>90=f₂(4;3) — whereas for each fixed level the linear analogue S₂(k;ℓ) agrees, for all large k, with a polynomial strictly increasing in ℓ; so the transferred parameter does not inherit the shape of its original. We also prove fᵣ(k;ℓ) ≥ fᵣ(k) for every level, with no search, and show the inequality is strict where it has been computed. Every theorem, proposition and lemma below is machine-checked in Lean 4.

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