Short proofs in combinatorics, probability and number theory II
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arXiv
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| Autores principales: | , , , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866914456054792192 |
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| author | Alexeev, Boris Putterman, Moe Sawhney, Mehtaab Sellke, Mark Valiant, Gregory |
| author_facet | Alexeev, Boris Putterman, Moe Sawhney, Mehtaab Sellke, Mark Valiant, Gregory |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_06609 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Short proofs in combinatorics, probability and number theory II Alexeev, Boris Putterman, Moe Sawhney, Mehtaab Sellke, Mark Valiant, Gregory Combinatorics Number Theory Probability 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. |
| title | Short proofs in combinatorics, probability and number theory II |
| topic | Combinatorics Number Theory Probability |
| url | https://arxiv.org/abs/2604.06609 |