Improved polynomial-time algorithms for detecting and recovering planted $Θ(\sqrt{n})$-cliques

In the planted clique problem, one observes either an Erdős--Rényi graph on $n$ vertices or such a graph with a clique added to $k = k(n)$ vertices, and seeks to detect or recover the clique. It is widely believed that $k = Θ(\sqrt{n})$ is the smallest clique size for which polynomial-time algorithm