Find largest subset of vertices in graph so that every pair of vertices have an edge between them
Unfortunately, the answer is no.

In graph theory, such a subset is called a clique. (Essentially, a graph is a set of nodes and a set of edges between those nodes, and a clique is a set of nodes in a graph such that every node in the set has an edge to every other node in the set.) What you want is to find the largest clique in a given graph.

This is called the Maximum Clique Problem, and it's NP-hard. The decision problem version, "is there a clique with size $k$?", is in fact NP-complete. So an efficient solution to the Maximum Clique Problem would prove that P=NP and earn you the $1,000,000 prize and eternal fame.