Maximal and Maximum Cliques

Solution 1:

A $k$-clique is just any collection of $k$ vertices in which every possible edge is present. Thus, you are right to say, for example, a 4-clique contains many different 3-cliques and 2-cliques (the latter being just an edge).

For your other question, you may be confusing the words "maximum" and "maximal". To appreciate the difference, consider a graph that is the disjoint union of a 3-clique and two 4-cliques (so the graph has three components).

Both of the 4-cliques are maximum-sized cliques in the graph, since they are the largest cliques you can find anywhere in the graph.

A clique is maximal if it cannot be made any larger in that particular graph. In our example, the three components are each maximal cliques. As you said earlier, the 4-cliques contain many 3-cliques within them. These "subcliques" are not maximal, since they can be made larger (in particular, they can be extended to the 4-clique containing them).

Notice that the 3-clique component is maximal (it cannot be made any larger with vertices in our graph) but not maximum (it is not a largest clique in the graph).

Solution 2:

A clique is, as you say, an induced subgraph in which every vertex is connected to every other vertex. Cliques may be contained in one another: in fact, every subgraph of a clique is necessarily itself a clique. Nothing wrong or problematic with that. So if you have a 4-clique, then each of its four subgraphs with three vertices are cliques as well.

No, not every graph has a unique maximum clique. For example, take two triangles and connect one vertex of one of them to another vertex of the other. No set of four vertices are mutually connected, so there are no 4-cliques in that graph; the two triangles are both maximal cliques, and there is no single "largest clique".

2-cliques make sense, but they are the trivial case of cliques (uninteresting).