Mathematics formula expression of finding cliques

Question

I am actually trying to write a math expression of finding a clique with size x. I can tell it in English, but somehow it is hard for me to translate it into a math expression.

In a graph, G = (V(G), E(G)). If we need to find a clique with size x means to check if there exists x vertices, $V_1, V_2, ..., V_x$, such that .....

and then I do not know how to write a math expression to tell such that every node in $\{V_1, V_2, ..., V_x\}$ must have a link to every other node in $\{V_1, V_2, ..., V_x\}$ and all links must also belong to E(G).

Answer 1

If $E(G)$ is the set of unordered pairs $\{V_a,V_b\}$ of vertices $V_a,V_b\in V(G)$ that are connected by an edge, then you could finish your sentence as follows:

... such that $\{V_i,V_j\}\in E(G)$ for all $i,j\in\{1,\ldots,x\}$ with $i\neq j$.

You might also want to emphasize that the vertices $V_1$, ..., $V_x$ should be distinct. You could also check

... if there exists $C\subset V(G)$ with $|C|=x$ and $\{a,b\}\in E(G)$ for all distinct $a,b\in C$.