an independent set of size $k$ in a finite undirected graph is a set of $k$ vertices of the graph, no two of which are connected by an edge. the structure of independent sets of size $k$ as $k$ varies is of interest in probability, statistical physics, combinatorics, and computer science. in 1987, alavi, malde, schwenk and erdos conjectured that the number of independent sets of size $k$ in a tree is a unimodal sequence (this number goes up and then it goes down), and this problem is still open. a variation on this question is: do the number of independent sets of size $k$ form a unimodal sequence for erdos-renyi random graphs, or random trees? by adapting an argument of coja-oghlan and efthymiou, we show unimodality for erdos-renyi random graphs, random bipartite graphs and random regular graphs (with high probability as the number of vertices in the graph goes to infinity, when the expected degree of a single vertex is large). the case of random trees remains open, as we can only show weak partial results there.