We study the lift-and-project rank of the stable set polytope of graphs with respect to the Lovász--Schrijver SDP operator $\text{LS}_+$ applied to the fractional stable set polytope. In particular, we show that for every positive integer $\ell$, the smallest possible graph with $\text{LS}_+$-rank $\ell$ contains $3\ell$ vertices. This result is sharp and settles a conjecture posed by Lipták and the second author in 2003, as well as answers a generalization of a problem posed by Knuth in 1994. We also show that for every positive integer $\ell$ there exists a vertex-transitive graph on at most $4\ell+12$ vertices with $\text{LS}_+$-rank at least $\ell$.

翻译：我们研究稳定集多面体在Lovász--Schrijver半定规划算子$\text{LS}_+$作用于分数稳定集多面体时的提升-投影秩。特别地，我们证明对每个正整数$\ell$，$\text{LS}_+$-秩为$\ell$的最小可能图包含$3\ell$个顶点。该结果是最优的，解决了Lipták与第二作者在2003年提出的猜想，并回答了Knuth在1994年提出的一个问题的推广。我们还证明对每个正整数$\ell$，存在一个顶点数不超过$4\ell+12$的顶点传递图，其$\text{LS}_+$-秩至少为$\ell$。