Restricted Schur multiplicity: exact minima and three-block minimisers

Abstract

Fix integers k ≥ 2 and 2 ≤ ℓ ≤ k, and call a solution of x₁ + … + xₖ = xₖ₊₁ in positive integers a solution of level ℓ when its k+1 entries take exactly ℓ+1 distinct values. Let m(k,ℓ;n) be the least number of monochromatic level-ℓ solutions inside [n] = {1,…,n}, minimised over all 2-colourings of [n], solutions being counted as multisets. The level ℓ is Gaiser's, who used it to index a threshold — the restricted generalized Schur number S₂(k;ℓ), the least n forcing one such solution — and who never counts; counting is the Schur-triple multiplicity problem of Graham–Rödl–Ruciński and Robertson–Zeilberger, which does not leave k = 2 and does not restrict the number of distinct values. We compute m(k,ℓ;n) exactly at 74 triples spread over seven pairs (k,ℓ) with k ≤ 5 and n ≤ 55; ten of them, the row k = ℓ = 2, reproduce Robertson–Zeilberger's own objective function and are carried as an external control. At every one of the 74 triples the minimum is attained by a colouring with at most three blocks, [1,a] ∪ (b,n] against (a,b]. We also prove that m(k,ℓ; ·) is non-decreasing; that it is positive exactly from n = S₂(k;ℓ) on, so that the multiplicity function recovers the threshold as its own zero set and re-proves S₂(3;2) = 15, S₂(3;3) = 24 and S₂(4;3) = 35 by a route that runs through no threshold search at all; and that a two-block colouring gives m(k,ℓ;n) = O((n/k)^(ℓ)) for every n. Every statement is machine-checked in Lean 4, except at the few places where we say otherwise; the lower bounds are replays of LRAT refutations of a cardinality-constrained encoding.

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