  We study the lift-and-project rank of the stable set polytopes of graphs with
respect to the Lov\'asz-Schrijver SDP operator $\text{LS}_+$. In particular, we
focus on a search for relatively small graphs with high $\text{LS}_+$-rank
(i.e., the least number of iterations of the $\text{LS}_+$ operator on the
fractional stable set polytope to compute the stable set polytope). We provide
families of graphs whose $\text{LS}_+$-rank is asymptotically a linear function
of its number of vertices, which is the best possible up to improvements in the
constant factor. This improves upon the previous best result in this direction
from 1999, which yielded graphs whose $\text{LS}_+$-rank only grew with the
square root of the number of vertices.
