  We consider the maximum independent set problem on graphs with maximum
degree~$d$. We show that the integrality gap of the Lov\'asz
$\vartheta$-function based SDP is $\widetilde{O}(d/\log^{3/2} d)$. This
improves on the previous best result of $\widetilde{O}(d/\log d)$, and almost
matches the integrality gap of $\widetilde{O}(d/\log^2 d)$ recently shown for
stronger SDPs, namely those obtained using poly-$(\log(d))$ levels of the
$SA^+$ semidefinite hierarchy. The improvement comes from an improved
Ramsey-theoretic bound on the independence number of $K_r$-free graphs for
large values of $r$.
  We also show how to obtain an algorithmic version of the above-mentioned
$SA^+$-based integrality gap result, via a coloring algorithm of Johansson. The
resulting approximation guarantee of $\widetilde{O}(d/\log^2 d)$ matches the
best unique-games-based hardness result up to lower-order poly-$(\log\log d)$
factors.
