Here's a simple and quick way to generate a planted clique in an Erdős–Rényi–Gilbert graph in #Mathematica. Below, such a random graph for n=60, k=15, p=1/2.

n = 60; k = 15; p = 1/2;
G = RandomGraph[BernoulliGraphDistribution[n, p]];
K = RandomSample[Range[n], k];
H = EdgeAdd[G, Map[UndirectedEdge[#[[1]], #[[2]]] &, Subsets[K, {2}]], EdgeStyle -> Map[UndirectedEdge[#[[1]], #[[2]]] -> Red &, Subsets[K, {2}]]];
H = Graph[H, VertexStyle -> Map[# -> Red &, K]];

(repost from an earlier tweet)