Short proofs in combinatorics, probability and number theory II

TL;DR

This paper presents five new proofs related to Erdős's questions in combinatorics, probability, and number theory, covering topics from planar point sets to prime number properties.

Contribution

It introduces novel proofs for five longstanding questions posed by Erdős, advancing understanding in multiple mathematical domains.

Findings

Proofs of properties of ordinary lines in planar point sets

Results on sequences with small exponential sums

Counterexample related to the Erdős–Turán discrepancy bound

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.