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arxiv: 2605.29011 · v1 · pith:R7STRRBYnew · submitted 2026-05-27 · 🧮 math.CO · math.NT

Subsequence Sums in Permutations

Collier Gaiser , Paul Horn This is my paper

Pith reviewed 2026-06-29 11:09 UTC · model grok-4.3

classification 🧮 math.CO math.NT
keywords permutationssubsequencesadditive subsequencesRamsey theorymonotone subsequencessum conditions
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The pith

For every k at least 3, every sufficiently large permutation of 1 through n contains a 2-additive subsequence of length k.

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

The paper shows that in any ordering of the integers from 1 to n, once n exceeds a threshold depending on k, there must exist a subsequence of length k whose elements sum to exactly twice the first element or twice the last element. This result applies arithmetic Ramsey theory ideas directly to permutations and supplies polynomial upper bounds on the smallest n that forces the property. It also determines that 18 is the exact threshold when the subsequence is required to be monotone, and it gives lower bounds on the total number of such subsequences that must appear. The same methods yield parallel existence statements when addition is replaced by multiplication or by taking reciprocals of the sums.

Core claim

For every integer k greater than or equal to 3 there exists an integer N such that every permutation of the set {1, 2, ..., n} with n greater than N contains a 2-additive subsequence of length exactly k. Polynomial bounds are proved for this N, the threshold for monotone 2-additive subsequences of length 3 is shown to be exactly 18, and quantitative lower bounds are obtained for the minimum number of 2-additive subsequences (of any length) and of monotone 2-additive triples that appear in every permutation.

What carries the argument

A 2-additive subsequence: an ordered tuple a1, a2, ..., ak whose sum equals twice a1 or twice ak.

If this is right

  • For each fixed k at least 3 the minimal n forcing a 2-additive k-subsequence in every permutation grows at most polynomially in k.
  • Every permutation of {1, 2, ..., 18} contains a monotone 2-additive subsequence of length 3, and there exist permutations of {1, 2, ..., 17} without one.
  • The same existence theorem holds when the sum condition is replaced by a product condition or by an inverse-sum condition.
  • Every permutation contains at least a positive polynomial number of 2-additive subsequences of unrestricted length.

Where Pith is reading between the lines

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

  • The polynomial bound on the threshold n may be improvable to a linear or even logarithmic function of k.
  • The same Ramsey-theoretic approach could be used to force 2-additive subsequences inside other structured families of sequences, such as those avoiding certain patterns.
  • Quantitative lower bounds on the number of such subsequences suggest that random permutations contain many more than the guaranteed minimum.

Load-bearing premise

Arithmetic Ramsey theory methods can be applied to locate these endpoint-sum properties inside every possible ordering of consecutive integers.

What would settle it

A single explicit permutation of {1, 2, ..., n} for some k greater than or equal to 3 and sufficiently large n that contains no 2-additive subsequence of length k.

read the original abstract

A sequence of positive integers $(a_1,a_2,\ldots,a_k)$ is called $\ell$-additive if $a_1+a_2+\cdots+a_k=\ell a_1$ or $\ell a_k$. In this paper, we prove that for all $k\geq3$, if $n$ is sufficiently large, then every permutation of $\{1,2,\ldots,n\}$ has a 2-additive subsequence of length $k$. We also provide polynomial bounds for the smallest $n$ such that every permutation of $\{1,2,\ldots,n\}$ has a 2-additive subsequence of length $k$. When only monotone subsequences are considered, we show that $18$ is the smallest $n$ such that every permutation of $\{1,2,\ldots,n\}$ has a monotone 2-additive subsequence of length three. Strong bounds are obtained for the minimum number of $\ell$-additive subsequences of any length, as well as monotone $2$-additive subsequences of length three. Using techniques in arithmetic Ramsey theory, we also show similar results for products and inverse sums.

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

1 major / 1 minor

Summary. The paper proves that for all k≥3 and sufficiently large n, every permutation of {1,2,…,n} contains a 2-additive subsequence of length k (sum equals 2·first or 2·last element). Polynomial bounds on the minimal such n are given. For monotone subsequences the exact threshold is shown to be 18 when k=3. Quantitative lower bounds on the minimum number of ℓ-additive subsequences (any length) and of monotone 2-additive subsequences of length 3 are obtained. Analogous existence and bound results are stated for products and inverse sums, all derived via arithmetic Ramsey theory.

Significance. If the central claims hold, the work supplies a concrete extension of arithmetic Ramsey theory to the setting of arbitrary linear orders on [n]. The polynomial bounds on the threshold n and the exact determination that 18 is minimal for the monotone k=3 case are explicit, falsifiable contributions. The lower bounds on the number of such subsequences add a quantitative dimension. Credit is due for obtaining these concrete numbers and for extending the method to products and inverse sums.

major comments (1)
  1. [proof of the main existence theorem] The central existence claim for arbitrary permutations rests on an adaptation of arithmetic Ramsey techniques that must locate solutions to a1+⋯+ak=2a1 or 2ak while respecting the given linear order. The manuscript invokes these techniques but supplies no explicit verification that density or uniformity hypotheses survive arbitrary reorderings (e.g., clustering of large or small values). This verification is load-bearing for the reduction from the unordered Ramsey statement to the ordered subsequence case and must appear in the proof of the main theorem.
minor comments (1)
  1. [abstract and quantitative results section] The abstract states that 'strong bounds' are obtained on the minimum number of subsequences but does not indicate their form (e.g., linear, quadratic). The main text should state the precise asymptotic or explicit expressions.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed report and for identifying a point that requires clarification in the proof of the main existence theorem. We address the comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: The central existence claim for arbitrary permutations rests on an adaptation of arithmetic Ramsey techniques that must locate solutions to a1+⋯+ak=2a1 or 2ak while respecting the given linear order. The manuscript invokes these techniques but supplies no explicit verification that density or uniformity hypotheses survive arbitrary reorderings (e.g., clustering of large or small values). This verification is load-bearing for the reduction from the unordered Ramsey statement to the ordered subsequence case and must appear in the proof of the main theorem.

    Authors: We agree that the manuscript invokes the arithmetic Ramsey results without an explicit verification step addressing preservation of density and uniformity conditions under arbitrary permutations. This is a valid observation, and the reduction step would benefit from such a verification to handle potential clustering of values. We will add a dedicated paragraph (or subsection) to the proof of the main theorem that explicitly confirms the hypotheses survive any reordering. The argument will note that the sum conditions are value-based (independent of positions) while the subsequence is extracted respecting the position order, with the polynomial bounds ensuring the required densities hold uniformly. revision: yes

Circularity Check

0 steps flagged

No circularity; proof applies external Ramsey techniques to permutations

full rationale

The paper states a direct existence proof for 2-additive subsequences in arbitrary permutations of [n] for large n, framed as an adaptation of arithmetic Ramsey theory methods. The abstract and description supply no equations or steps that reduce the claimed result to a self-definition, a fitted input renamed as prediction, or a load-bearing self-citation chain. The derivation is presented as self-contained against the external body of Ramsey-theoretic results, with no quoted reduction showing the target statement equivalent to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on the applicability of arithmetic Ramsey theory to the 2-additive condition inside permutations; no free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Techniques from arithmetic Ramsey theory suffice to guarantee the existence of 2-additive subsequences in arbitrary permutations of [n] for large n.
    Explicitly invoked in the abstract for the main results and the extensions to products and inverse sums.

pith-pipeline@v0.9.1-grok · 5721 in / 1161 out tokens · 35185 ms · 2026-06-29T11:09:20.453223+00:00 · methodology

discussion (0)

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Reference graph

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