PapersMulti-source
ChatGPT-6 Astra 找到偏序集维度定理的完整证明
On dimension of posets excluding a standard example
Why we picked it
OpenAI 的 ChatGPT-6 Astra 找到组合数学里偏序集维度的完整证明，还带上下界，数学圈该看看。
论文证明：对每个整数 d≥3，存在 α<1，使得不含 d 阶标准例的 n 元偏序集的最大维度为 O_d(n^α)。当 d=3 时给出上界 O(n^{2/3}log n)，并构造了维度为 Ω(n^{1/3}/log n) 的 S3-free 偏序集族。论文声明这些证明由 OpenAI 的 ChatGPT-6 Astra 找到。
Original · arXiv: OpenAI
On dimension of posets excluding a standard example
We prove that for every integer $d \geq 3$, there exists $α<1$ such that the maximum dimension of an $n$-element poset containing no standard example of order $d$ as an induced subposet is $\mathcal{O}_d(n^α)$. For $d=3$, we obtain an upper bound $\mathcal{O}(n^{2/3}\log n)$, while we also construct a family of $S_3$-free posets with dimension $Ω(n^{1/3}/\log n)$. The proofs were found by OpenAI's ChatGPT-6 Astra.
