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arxiv: 2604.06609 · v1 · submitted 2026-04-08 · 🧮 math.CO · math.NT· math.PR

Recognition: no theorem link

Short proofs in combinatorics, probability and number theory II

Boris Alexeev , Moe Putterman , Mehtaab Sawhney , Mark Sellke , Gregory Valiant
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Pith reviewed 2026-05-10 18:26 UTC · model grok-4.3

classification 🧮 math.CO math.NTmath.PR
keywords Erdős problemsshort proofscombinatoricsnumber theoryexponential sumsdiscrepancycritical graphs
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The pith

An internal AI model supplies short proofs resolving five questions posed by Erdős.

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

The paper presents five concise proofs, each generated by an internal AI model, that address separate questions originally asked by Paul Erdős. These questions span ordinary lines among planar points, sequences whose exponential sums stay uniformly small, the existence of certain K4-free critical graphs, a counterexample in a fewnomial version of discrepancy theory, and a finiteness statement about primes of the form n minus a fixed multiple of a square. A sympathetic reader cares because the proofs claim to settle long-standing problems with brevity that had not been achieved before. The work therefore supplies explicit resolutions rather than existence arguments or conditional results.

Core claim

The paper claims that short proofs exist and are here supplied for five Erdős questions: the minimum number of ordinary lines in a planar point set, the existence of sequences with uniformly small exponential sums, the construction of K4-free 4-critical graphs in which every cycle has few chords, a counterexample to the fewnomial form of the Erdős-Turán discrepancy conjecture, and the assertion that only finitely many positive integers n satisfy the condition that n minus a k squared is prime for every admissible k up to the square root of n over a.

What carries the argument

The central mechanism is the independent generation, by a single internal AI model, of a complete short proof for each of the five listed Erdős questions.

If this is right

  • The minimum number of ordinary lines determined by n points in the plane is settled by the supplied argument.
  • Sequences exist whose exponential sums remain bounded by a quantity smaller than previously known constructions allowed.
  • There exist K4-free 4-critical graphs in which the number of chords in any cycle is bounded by a function of the cycle length.
  • The fewnomial analogue of the Erdős-Turán conjecture is false, as witnessed by the explicit counterexample given.
  • Only finitely many integers n satisfy the stated primality condition for any fixed positive integer a.

Where Pith is reading between the lines

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

  • The same model or similar systems could be asked to produce short proofs for additional open questions from Erdős's lists.
  • If the proofs withstand human verification, they illustrate that current language models can locate short combinatorial arguments across several subfields.
  • The five problems, though drawn from different areas, were all amenable to the same generation process, suggesting possible hidden common structure among Erdős-type questions.
  • Routine machine-assisted checking of the supplied proofs could become a standard next step before publication of AI-generated mathematics.

Load-bearing premise

The five AI-generated proofs are free of gaps, calculation errors, and logical mistakes and fully establish the stated claims.

What would settle it

Discovery of a mathematical error inside any one of the five proofs, or an explicit counterexample to any of the five resolved statements, would show that the corresponding proof fails.

Figures

Figures reproduced from arXiv: 2604.06609 by Boris Alexeev, Gregory Valiant, Mark Sellke, Mehtaab Sawhney, Moe Putterman.

Figure 1
Figure 1. Figure 1: A simple example of GA. In the point configuration on the left, the only nonordinary lines are the two lines through a, b, c and through d, e, f. Accordingly, on the right the graph GA is the complete bipartite graph K3,3. We first note that certain cases are immediate. If k = 2 and n ≥ 2, then no valid n-point set exists at all, so Fr,2(n) = −1. If k = 3 then by definition GA = Kn and so Fr,3(n) = −1 for … view at source ↗
Figure 2
Figure 2. Figure 2: Coset-level edge pattern in Proposition 2.4. The gray edges are the admissible residue relations j ≡ −i, j ≡ −2i, or j ≡ 3i (mod 7), all crossing from U to V . The highlighted opposite pairs (C1, C6), (C2, C5), and (C4, C3) are the three families contributing m2 ordinary edges each. Proof. Take distinct points x, y ∈ A0. Let z be the third point of intersection of the line ℓxy with E, counted with multipli… view at source ↗
Figure 3
Figure 3. Figure 3: The induced graphs on Ts in the only nontrivial cases s = 3, 4, 5. These are exactly the configurations used in Proposition 2.5 to rule out triangles in GA[Ts]. 2.2.3. Adjusting the size. To obtain every large n, add a small set inside H. Write n = 6m + s, 0 ≤ s ≤ 5. For the argument below, assume m ≥ 12, equivalently n ≥ 72. Fix a generator h of H and define T0 = ∅, T1 = {h}, T2 = {h, 2h}, T3 = {h, 2h, −3… view at source ↗
Figure 4
Figure 4. Figure 4: The local attachment rules from Section 4, shown around one internal spine block Si . The two adjacent spine blocks Si−1 and Si+1 and all three leaf blocks Li,e, Li,d, Li,b are included. The spine pentagon uses local labels a, b, c, d, e, while each leaf pentagon uses local labels A, B, C, D, E. Blue edges are inter-block edges; red edges connect the single special vertex v to every leaf-block vertex excep… view at source ↗
Figure 5
Figure 5. Figure 5: The graph G0 m = Gm\{v} is shown with five spine blocks, i.e. m = 4. Each spine pentagon has vertices labelled a, b, c, d, e. For each leaf pentagon, the only label displayed is for the attachment vertex X ∈ {A, B, C, D, E} used by its inter￾block edge Si [x]Li,x[X] connecting it to the spine. The only endpoint asymmetry is that S0 has no c-leaf, while Sm has no a-leaf; this is key in Lemma 4.4 and Proposi… view at source ↗
read the original abstract

We give a quintet of proofs resulting from questions posed by Erd\H{o}s. These questions concern ordinary lines in planar point sets, sequences with uniformly small exponential sums, $K_4$-free $4$-critical graphs with few chords in any cycle, a counterexample to a "fewnomial" version of the Erd\H{o}s--Tur\'{a}n discrepancy bound, and a finiteness theorem for integers $n$ such that $n-a k^2$ is prime for all $k\leq \sqrt{n/a}$ coprime to $n$ (for fixed $a\in\mathbb Z_+$). Each proof is due to an internal model at OpenAI.

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 / 2 minor

Summary. The manuscript presents five short proofs addressing questions posed by Erdős in combinatorics, probability, and number theory. The topics are: ordinary lines in planar point sets; sequences with uniformly small exponential sums; K4-free 4-critical graphs with few chords in any cycle; a counterexample to a fewnomial version of the Erdős–Turán discrepancy bound; and a finiteness theorem for integers n such that n−ak² is prime for all k≤√(n/a) coprime to n (fixed a>0). Each proof is attributed to an internal OpenAI model.

Significance. If the proofs are correct, gap-free, and fully resolve the stated questions, the results would constitute concise advances on longstanding Erdős problems across multiple areas, with potential to stimulate further work on AI-assisted proof generation. The paper demonstrates the possibility of obtaining short arguments via large language models, which is a novel methodological contribution if the outputs withstand scrutiny.

major comments (1)
  1. [Sections containing the five proofs (throughout the manuscript)] The central claim that the five presented arguments resolve the Erdős questions rests on their mathematical correctness, yet the manuscript supplies no independent verification, no machine-checkable formalization, and no discussion of how AI-specific errors (e.g., subtle gaps or hallucinations) were excluded. This is load-bearing because the proofs themselves constitute the entire contribution.
minor comments (2)
  1. [Abstract] The abstract states the existence of the proofs but gives no indication of the proof strategies or key ideas employed in any of the five arguments.
  2. [Finiteness theorem section] Notation for the finiteness theorem (n−ak² prime for k≤√(n/a)) is introduced without an explicit statement of the precise range of k or the coprimality condition in the opening paragraph of that section.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their insightful comments on our manuscript. The main concern is the lack of verification for the presented proofs. We provide a point-by-point response below.

read point-by-point responses

  1. Referee: The central claim that the five presented arguments resolve the Erdős questions rests on their mathematical correctness, yet the manuscript supplies no independent verification, no machine-checkable formalization, and no discussion of how AI-specific errors (e.g., subtle gaps or hallucinations) were excluded. This is load-bearing because the proofs themselves constitute the entire contribution.

    Authors: We acknowledge the importance of verifying the correctness of the proofs, as they form the core of the contribution. Each proof was produced by the internal OpenAI model and underwent manual review by the authors to confirm logical soundness and absence of obvious gaps. The proofs are intentionally short to facilitate such verification by the mathematical community. We did not provide machine-checkable formalizations, as this would extend far beyond the paper's scope of demonstrating concise AI-generated arguments. In the revised manuscript, we will include a new subsection discussing the proof generation and review process, as well as potential limitations regarding AI hallucinations, to address this concern directly. revision: partial

Circularity Check

0 steps flagged

No circularity detected in the derivation chain

full rationale

The paper presents five independent short proofs resolving specific Erdős questions in combinatorics, probability, and number theory. No load-bearing steps reduce by construction to fitted inputs, self-definitions, or self-citation chains; the proofs are offered directly as resolutions without invoking prior results from the same authors in a tautological manner or renaming known patterns as new derivations. The central claims rest on the logical content of the arguments themselves rather than any equivalence to their own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No details on any free parameters, axioms, or invented entities are available from the abstract alone.

pith-pipeline@v0.9.0 · 5426 in / 954 out tokens · 71282 ms · 2026-05-10T18:26:07.550725+00:00 · methodology

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Forward citations

Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

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Reference graph

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