On dimension of posets excluding a standard example

We prove that for every integer $d\ge 3$, there exists $\alpha <1$ such that the maximum dimension of an $n$-element poset containing no standard example of order $d$ as an induced subposet is ${\mathcal{O}}_d(n^{\alpha })$. For $d=3$, we obtain an upper bound $\mathcal{O}(n^{2\mathrm{/}3}\log n)$, while we also construct a family of $S_3$-free posets with dimension $\mathrm{\Omega }(n^{1\mathrm{/}3}\mathrm{/}\log n)$. The proofs were found by OpenAI’s ChatGPT-6 Astra.