Short proofs in combinatorics, probability and number theory II

Abstract: We give a quintet of proofs resulting from questions posed by Erdős. 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 Erdős--Turán 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.

• The paper presents concise AI-assisted proof strategies that resolve five longstanding Erdős problems in combinatorics, probability, and number theory.
• It employs innovative constructions such as elliptic curve configurations, randomized dyadic decompositions, and extremal graph designs to settle conjectures with sharp counterexamples.
• The work bridges algebraic geometry, probabilistic methods, and number theory, demonstrating AI’s potential to generate novel and valid mathematical insights.

Authoritative Summary of "Short proofs in combinatorics, probability and number theory II" (2604.06609)

Overview

This paper presents concise solutions to five problems posed by Erdős, spanning combinatorics, probability, and number theory. The authors utilize advanced algebraic, analytic, and probabilistic constructions to settle conjectures and answer open questions regarding extremal configurations, exponential sum bounds, color-critical graphs, polynomial root distribution, and number-theoretic finiteness phenomena. Notably, all proof strategies originate from internal AI model output, with post-processing performed by human experts.

Ordinary Lines and the Erdős Conjecture in Combinatorial Geometry

The first result addresses the growth rate of ordinary lines in planar point sets, under the constraints of avoiding large collinear subsets and forbidden cliques in the ordinary-line graph. Erdős conjectured that restricting $k$-point collinearities and $K_r$ cliques in the ordinary-line graph could force the maximal number of ordinary lines, $F_{r,k}(n)$, to be subquadratic or even linear in $n$. The authors disprove this for $k \ge 4$, $r \ge 3$, constructing explicit $n$-point sets (derived from large cyclic subgroups on real elliptic curves) with no four collinear points, triangle-free (bipartite) ordinary-line graphs, and at least $n^2/12 - O(n)$ ordinary lines.

This construction leverages the group law on a connected nonsingular elliptic curve, partitioning points into residue classes and exploiting the collinearity structure to maximize ordinary lines while forbidding small cliques. This approach both tightens classical bounds and extends recent combinatorial geometry advances.

Strong result: The existence of configurations with quadratically many ordinary lines despite stringent forbidden configurations refutes conjectures of subquadratic growth, demonstrating that geometric and combinatorial constraints do not substantially restrict ordinary line proliferation beyond the known Turán bound.

Small Uniform Exponential Sums: Nearly-Optimal Sequences

For the problem of constructing sequences with uniformly small complex exponential sums—an area connecting harmonic analysis to combinatorial number theory—the authors provide a construction based on randomized dyadic decompositions and scrambled binary expansions. They achieve a sequence $(x_n)$ such that for all $k \in \mathbb{N}$,

$K_r$0

This matches, up to $K_r$1 factors, the lower bound established by Clunie and represents a sharp improvement over previous deterministic constructions with linear-in-$K_r$2 bounds.

The proof utilizes multi-scale dyadic block decompositions, probabilistic argumentation with conditioned independence, and Bernstein’s inequality to control the maximum over all subintervals and all exponential frequencies.

Implication: This provides an explicit method for constructing real sequences with near-optimal small exponential sum growth, resolving an Erdős question and formalizing a probabilistic improvement over van der Corput and Clunie-type sequences.

Extremal $K_r$3-Free 4-Critical Graphs with Bounded Cycle Chords

The third result focuses on the existence of $K_r$4-free graphs with chromatic number $K_r$5, all small subgraphs 3-colorable, yet all cycles (including odd cycles) having a bounded number of chords. This negatively answers an Erdős conjecture suggesting a tradeoff between local colorability and global chord structure.

The authors provide an explicit infinite family of such graphs, each with $K_r$6, every proper subgraph $K_r$7-degenerate (and hence $K_r$8-colorable), and with maximal cycle chord number at most $K_r$9.

The construction intricately arranges pentagonal blocks in a certain caterpillar-like structure, attaching special vertices and inter-block connections to ensure both the critical colorability and chord bounds.

Contradictory claim: This disproves the conjectured necessity for high chord count in cycles of locally 3-colorable, globally 4-chromatic, $F_{r,k}(n)$0-free graphs, showing that local sparsity and global coloring complexity can coexist with bounded chordality.

Counterexample to the Sparse Erdős-Turán Discrepancy Bound

The fourth result produces an explicit family of lacunary polynomials violating a natural sparse analogue of the classical Erdős-Turán root discrepancy theorem. The proposed improvement would replace the degree $F_{r,k}(n)$1 in the discrepancy bound by the number of nonzero coefficients $F_{r,k}(n)$2, aiming for

$F_{r,k}(n)$3

for all intervals, where $F_{r,k}(n)$4 measures coefficient sizes. The authors construct polynomials (via Vandermonde determinants and large root multiplicities) with bounded $F_{r,k}(n)$5, $F_{r,k}(n)$6, but an interval containing $F_{r,k}(n)$7 roots—exceeding $F_{r,k}(n)$8 discrepancy—for all large $F_{r,k}(n)$9.

Theoretical implication: This answers negatively to the sparse Erdős-Turán strengthening conjectured by Erdős, showing that high multiplicity and sparse support can defeat expected distribution uniformity, and that Hayman’s prior bound of discrepancy by $n$0 is essentially tight.

Finiteness Theorem for Primes of the Form $n$1

The final contribution is a finiteness theorem: for fixed $n$2, only finitely many integers $n$3 exist such that $n$4 is a prime for all $n$5 coprime to $n$6. This resolves Problem 1141 of Erdős (and its extensions) by reducing the problem to quadratic residue conditions modulo primes, and applying deep results about the distribution of small prime quadratic residues (Pollack’s theorem).

The argument splits cases by the structure of $n$7, employing sieve methods, quadratic characters, quantitative versions of the Chebotarev density theorem, and explicit lower bounds on available solutions, contradicting primitivity if $n$8 is large.

Practical implication: Even under relaxed coprimality conditions, the sequence of such $n$9 is finite and effectively unbounded above.

Role of AI in Proof Generation

Each argument was generated by an internal AI model, with partial success in attempts using ChatGPT variants. Human authors verified and polished the outputs, adding simplifications where appropriate but retaining the essential AI-generated strategies.

Conclusion

The collection unifies five distinct, previously unresolved Erdős-inspired problems through compact, technically sophisticated solutions across combinatorics, probability, and analytic number theory. By leveraging algebraic geometry (for planar configurations), probabilistic block decompositions (for exponential sum control), extremal coloring and degeneracy constructions (for graph theory), explicit root-multiplicity interpolation (for polynomials), and analytic number theory (for quadratic residues), the paper decisively settles multiple conjectures and opens pathways for further exploration into AI-assisted research and the structural possibilities for extremal objects in discrete mathematics and analytic number theory.

Potential future directions include: automating the generation of extremal combinatorial examples, improved randomized constructions in discrepancy theory, and extending the interplay between algebraic, probabilistic, and number-theoretic techniques in finding counterexamples and extremal configurations. The demonstrated effectiveness of advanced AI models in producing such nontrivial arguments suggests a growing role for computational and AI-driven intuition in mathematical discovery.