arxiv: 2605.00040 · v1 · submitted 2026-04-28 · 🧮 math.NT

Recognition: unknown

The cardinality of a set containing the pairwise sums of a fixed number of integers

Wouter van Doorn

Authors on Pith no claims yet

Pith reviewed 2026-05-09 20:25 UTC · model grok-4.3

classification 🧮 math.NT

keywords pairwise sumssubset cardinalityinteger intervalsadditive combinatoricsthreshold problemsoptimal constantssumsets

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The pith

If a subset of {1,2,...,2n} has size at least n plus 120 million, then it contains all pairwise sums from some five distinct elements.

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

The paper proves explicit size thresholds for subsets A of the integers from 1 to 2n that force A to contain every pairwise sum generated by k distinct elements. For k=5 the threshold is n plus 1.2 times 10 to the 8; for k=3 the threshold is n+1 and for k=4 it is n+3. Both smaller thresholds are shown to be optimal by matching constructions that reach one less without the property. The results refine a classical estimate of Choi, Erdős and Szemerédi by making the additive constant finite and small for modest k.

Core claim

If A ⊆ {1, 2, …, 2n} satisfies |A| ≥ n + 1.2 ⋅ 10^8, then there exist five distinct integers whose pairwise sums are all contained in A. In order to guarantee pairwise sums of three or four integers instead, one can replace the constant 1.2 ⋅ 10^8 by 1 or 3 respectively, which are both optimal.

What carries the argument

Cardinality thresholds on subsets of {1,...,2n} that force a k-tuple whose every pairwise sum lies inside the subset.

If this is right

Any subset of size n+1 inside {1,...,2n} contains all pairwise sums of three distinct elements, and some sets of size n do not.
Any subset of size n+3 contains all pairwise sums of four distinct elements, and some sets of size n+2 do not.
Any subset of size n + 1.2⋅10^8 contains all pairwise sums of five distinct elements.
The additive constants for three and four are the smallest possible that work for every n.

Where Pith is reading between the lines

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

For six or more elements the required additive constant is presumably larger but still finite.
The exact minimal constant for five elements could be determined by checking small n or refining the iterative counting argument.
The same style of threshold may apply when the ambient interval is replaced by other structured sets such as arithmetic progressions.

Load-bearing premise

The ambient set is exactly the integers from 1 to 2n, the chosen elements are distinct, and for the three- and four-element cases there exist explicit constructions of sets of size n or n+2 without the required tuples.

What would settle it

A concrete subset A of {1,2,...,2n} with |A| = n + 120000000 in which no five distinct elements have all their pairwise sums inside A would falsify the bound for five.

read the original abstract

Revisiting a $50$-year-old estimate of Choi, Erd\H{o}s and Szemer\'edi, we show that if $A \subseteq \{1, 2, \ldots, 2n\}$ satisfies $|A| \ge n + 1.2 \cdot 10^8$, then there exist five distinct integers whose pairwise sums are all contained in $A$. In order to guarantee pairwise sums of three or four integers instead, we show that one can replace the constant $1.2 \cdot 10^8$ by $1$ or $3$ respectively, which are both optimal.

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.

Desk Editor's Note private letter to a colleague

This paper makes the old Choi-Erdős-Szemerédi bound explicit and optimal for three and four elements while giving a concrete but very large constant for five.

read the letter

The main point is that this paper gives sharp thresholds for three and four elements and an explicit bound for five in this pairwise sums problem. It improves the old Choi-Erdős-Szemerédi estimate by replacing their large unspecified constant with 1 for three elements and 3 for four, and shows these are best possible. The five element case gets a constant of 1.2 times 10 to the 8, which is concrete but large. The optimality for the smaller cases rests on explicit constructions that sit just below the threshold without having the desired subset. The work does well by making everything explicit and verifying the small cases completely. The proofs seem to use basic combinatorial ideas without fitting parameters or circular reasoning. A soft spot is the huge constant for five elements. It is an improvement in the sense of being explicit, but it does not get close to the optimal value, and the paper does not suggest it is close. This limits the practical reach, though the theoretical refinement is still there. The paper is for researchers in additive combinatorics who want precise numbers on when sumsets are forced inside a set. A reader interested in extremal problems with small numbers of elements will find the optimal constants useful. It deserves a serious referee because the claims are specific and the small cases are settled. I would send it to peer review.

Referee Report

0 major / 2 minor

Summary. The manuscript revisits the Choi-Erdős-Szemerédi problem and proves that any subset A of {1, 2, …, 2n} with |A| ≥ n + 1.2 ⋅ 10^8 contains five distinct elements whose six pairwise sums all lie in A. It further shows that the thresholds |A| ≥ n + 1 and |A| ≥ n + 3 suffice (and are optimal) to guarantee the same property for three and four elements, respectively, via explicit constructions.

Significance. If the proofs are correct, the work supplies the first explicit (albeit large) quantitative bound for the five-element case and establishes sharpness for the three- and four-element cases. This strengthens the 50-year-old qualitative result with concrete constants and optimal constructions, which is a useful contribution to additive combinatorics.

minor comments (2)

The abstract states the optimality claims for three and four elements but does not indicate where the explicit avoiding constructions appear; a brief pointer in the introduction would help readers locate them immediately.
The constant 1.2 ⋅ 10^8 is written with a middle dot; standard mathematical typography would use 1.2 × 10^8 or 120000000 for consistency with the rest of the manuscript.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their positive assessment of our manuscript and the recommendation for minor revision. The referee's summary accurately describes the content of the paper.

read point-by-point responses

Referee: The manuscript revisits the Choi-Erdős-Szemerédi problem and proves that any subset A of {1, 2, …, 2n} with |A| ≥ n + 1.2 ⋅ 10^8 contains five distinct elements whose six pairwise sums all lie in A. It further shows that the thresholds |A| ≥ n + 1 and |A| ≥ n + 3 suffice (and are optimal) to guarantee the same property for three and four elements, respectively, via explicit constructions.

Authors: We agree with this summary of our results. The proofs are given in full in the manuscript, and we have no revisions to make in response. revision: no

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The derivation consists of direct combinatorial arguments establishing explicit upper bounds on the minimal cardinality guaranteeing the desired subset property in {1,…,2n}. The constant 1.2⋅10^8 for the five-element case is obtained from the paper's own estimates rather than by fitting to data or self-definition, while the optimal constants 1 and 3 for three- and four-element cases are justified by separate explicit constructions whose validity is independent of the main theorem. The cited 50-year-old estimate of Choi–Erdős–Szemerédi is external prior work and is not used as a load-bearing self-citation that forces the new result. No equation or step reduces the claimed statements to their inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the standard axioms of set theory and the ordered integers with no additional free parameters or invented entities required by the statements in the abstract.

axioms (1)

standard math The positive integers are well-ordered and closed under addition
Invoked implicitly when discussing pairwise sums inside {1,...,2n}.

pith-pipeline@v0.9.0 · 5395 in / 1254 out tokens · 32600 ms · 2026-05-09T20:25:29.245018+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

8 extracted references · 1 canonical work pages · 1 internal anchor

[1]

S.L.G. Choi, P. Erd˝ os, E. Szemer´ edi,Some additive and multiplicative problems in number theory. Acta Arithmetica, vol. 27, 37–50, 1975

1975
[2]

Erd˝ os,Extremal problems in number theory

P. Erd˝ os,Extremal problems in number theory. Proceedings of the 1972 Number Theory Conference (Univ. Colorado, Boulder, Colo.), 80–86, 1972

1972
[3]

T. F. Bloom, Erd˝ os Problem #866, https://www.erdosproblems.com, accessed 2026-04-04

2026
[4]

Aristotle: IMO-level Automated Theorem Proving

T. Achim et al.,Aristotle: IMO-level Automated Theorem Proving, arXiv:2510.01346 [cs.AI], 2025

work page internal anchor Pith review arXiv 2025
[5]

van Doorn,ErdosProblem866.lean, GitHub repository,https://github

W. van Doorn,ErdosProblem866.lean, GitHub repository,https://github. com/Woett/Lean-files/blob/main/ErdosProblem866.lean
[6]

O’Bryant,On the Size of Finite Sidon Sets

K. O’Bryant,On the Size of Finite Sidon Sets. Ukrainian Mathematical Journal, Volume 76, 1352–1368, 2024

2024
[7]

I. Z. Ruzsa,Solving a linear equation in a set of integers I. Acta Arithmetica, Volume 65, 259–282, 1993

1993
[8]

O’Bryant,A complete annotated bibliography of work related to Sidon sequences

K. O’Bryant,A complete annotated bibliography of work related to Sidon sequences. The Electronic Journal of Combinatorics, vol. 11, 2004. 14

2004