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arxiv: 2410.22024 · v2 · submitted 2024-10-29 · 🧮 math.CO · math.NT

An Unsure Note on an Un-Schur Problem

Olaf Parczyk , Christoph Spiegel This is my paper

Pith reviewed 2026-05-23 18:51 UTC · model grok-4.3

classification 🧮 math.CO math.NT
keywords Schur triplesrainbow coloringanti-Ramsey3-coloringintegersadditive combinatoricsSchur problem
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The pith

The maximum fraction of rainbow Schur triples in any 3-coloring of the first n integers lies between 0.4 and 0.66364.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper introduces an anti-Ramsey variant of the classic Schur triple problem. It shows that for large n, there are 3-colorings of [n] making at least 0.4 of the Schur triples rainbow, but no coloring makes more than 0.66364 of them rainbow. The lower bound is achieved by explicit constructions, and the upper bound follows from analytic arguments on color class sizes. The authors conjecture that the lower bound of 0.4 is asymptotically tight. This mirrors the settled problem of maximum rainbow triangles in 3-edge-colored complete graphs.

Core claim

Graham, Rödl, and Rucinski posed the minimum monochromatic Schur triples in 2-colorings, which was solved. Here the anti-Ramsey version for 3-colorings is considered, proving that the maximum fraction of rainbow Schur triples is at least 0.4 and at most 0.66364, with the conjecture that 0.4 is the precise value.

What carries the argument

Counting the fraction of Schur triples (solutions to x + y = z) that receive three distinct colors in a 3-coloring of [n].

If this is right

  • The fraction of rainbow Schur triples can reach at least 40 percent.
  • No 3-coloring exceeds roughly 66.364 percent rainbow Schur triples.
  • The asymptotic maximum is conjectured to be exactly 0.4.
  • Similar questions can be posed for other numbers of colors or different equations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Methods from extremal graph theory used for rainbow triangles might adapt to this setting for tighter bounds.
  • Verification on finite n could reveal how quickly the bounds are approached.
  • Generalizations to k-colorings or other additive configurations may follow similar patterns.
  • The connection to the Erdős-Sós problem suggests a possible link between additive and graph-theoretic rainbow problems.

Load-bearing premise

The constructions and upper bound proofs extend uniformly to arbitrarily large n without hidden dependencies on small-scale structure.

What would settle it

An explicit 3-coloring of [n] for large n with a rainbow Schur fraction exceeding 0.4 would falsify the conjecture, or one below 0.4 would falsify the lower bound.

Figures

Figures reproduced from arXiv: 2410.22024 by Christoph Spiegel, Olaf Parczyk.

Figure 1
Figure 1. Figure 1: Region of feasible α and β for fixed γ0 = 0.077102 with the first bound from Equa￾tion (15) in blue and the second bound from Equation (16) in green. The optimal point is in red with the corresponding contour line for the objective from Equation (13). The black dot corresponds to our lower bound construction. Second bound. Since z0 was again chosen maximal in Equation (11), we have Xn z=1 r(z) (n + 1 − z) … view at source ↗
read the original abstract

Graham, R\"odl, and Ruci\'nski originally posed the problem of determining the minimum number of monochromatic Schur triples that must appear in any 2-coloring of the first $n$ integers. This question was subsequently resolved independently by Datskovsky, Schoen, and Robertson and Zeilberger. Here we suggest studying a natural anti-Ramsey variant of this question and establish the first non-trivial bounds by proving that the maximum fraction of Schur triples that can be rainbow in a given $3$-coloring of the first $n$ integers is at least $0.4$ and at most $0.66364$. We conjecture the lower bound to be tight. This question is also motivated by a famous analogous problem in graph theory due to Erd\H{o}s and S\'os regarding the maximum number of rainbow triangles in any $3$-coloring of $K_n$, which was settled by Balogh et al.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript introduces an anti-Ramsey variant of the classical Schur triple problem, asking for the maximum asymptotic fraction of rainbow Schur triples achievable in any 3-coloring of [n]. It claims to prove that this fraction lies between 0.4 (via explicit construction) and 0.66364 (via an upper-bound argument), and conjectures that the lower bound is tight. The work is motivated by the resolution of the minimum-monochromatic-Schur-triples problem and by the analogous Erdős–Sós question on rainbow triangles in 3-edge-colored K_n.

Significance. If the claimed proofs hold, the paper supplies the first non-trivial bounds on a natural new problem in additive anti-Ramsey theory. The explicit lower-bound construction and the uniform upper-bound argument constitute concrete, falsifiable progress, while the conjecture supplies a precise target for subsequent work. The parallel drawn to the graph-theoretic result settled by Balogh et al. is apt and helps situate the contribution.

minor comments (3)
  1. The numerical upper bound 0.66364 is stated to five decimal places; a short remark on whether it arises from an exact closed-form expression, a linear-programming relaxation, or a finite-n optimization would improve transparency.
  2. The abstract asserts the existence of an explicit construction achieving the 0.4 lower bound; a brief description of the coloring (or a reference to the relevant section) would allow readers to verify the asymptotic density without reconstructing the argument.
  3. Consider adding a short concluding section that lists concrete open questions beyond the main conjecture, e.g., the behavior under more than three colors or for other additive configurations.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive evaluation of the manuscript, for situating the work within additive anti-Ramsey theory, and for recommending minor revision. We appreciate the acknowledgment that the explicit construction and uniform upper bound constitute concrete progress on this new problem.

Circularity Check

0 steps flagged

No significant circularity; new problem with independent bounds

full rationale

The paper introduces a new anti-Ramsey variant on rainbow Schur triples in 3-colorings and states explicit asymptotic bounds (0.4 lower via construction, 0.66364 upper) plus a conjecture. No self-definitional equations, fitted parameters renamed as predictions, or load-bearing self-citations appear in the derivation chain. The bounds rest on explicit constructions and uniform upper-bound arguments that are presented as independent of the target result itself. The problem formulation and motivation from prior external work (Graham-Rödl-Ruciński, Balogh et al.) do not reduce the central claims to the paper's own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no explicit free parameters, axioms, or invented entities; the problem is posed in standard terms of Ramsey theory.

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Reference graph

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