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Now there's a matching of size $s$ in $G$ if and only if there's a clique of size $s$ in $G'$. … In particular, there's a perfect bipartite matching for $G$ if and only if there is a clique of size $|V|/2$ in $G'$. …

It is at least as hard as the $k$-clique problem. Suppose we have a graph and we want to know whether it contains a $k$-clique. … Now the graph has a $k$-clique if and only if the answer to your problem is yes. But testing for a $k$-clique is NP-complete, so it follows that your problem is NP-complete too. …

Here is an algorithm with $O(n^{2+\lg k})$ running time. It's not polynomial in $k$ and $n$, but for fixed $n$ it is polynomial in $k$, and for fixed $k$ it is polynomial in $n$. Here is the algorit …

In particular, testing whether there exists a clique of size $\ge k$ is as hard (under worst-case complexity) as finding the size of the maximum clique. … (The reduction: if you had an algorithm for testing the existence of a clique, then you could use binary search on $k$ to find the size of the maximum clique.) …

The Clique problem is NP-complete. If a NP-complete problem was in co-NP, then we would have NP = co-NP. It is an open problem whether NP = co-NP, but most complexity theorists expect that NP ! … = co-NP -- which would imply that Clique is not in co-NP. …

Introduce zero-or-one variables $x_{i,v}$, $y_{i,u,v,w}$, where $x_{i,v}=1$ means that vertex $v$ is contained in partition $P_i$, and $y_{i,u,v,w}=1$ means that $u,v,w$ are a 3-clique in $P_i$. … (Only introduce variables $y_{i,u,v,w}$ for triples of vertices $u,v,w$ that are a 3-clique in the original graph $G$; all others are effectively hardcoded at 0.) …

The problem is NP-hard, so you shouldn't expect any efficient algorithm that will always work. You can look for heuristics, or approximation algorithms, or sometimes-efficient algorithms. If I had to …

No, you couldn't say that, because it's not true. A helpful method is to prove all your claims. Don't just guess -- try to find a proof. If you're struggling to find a proof, the first thing to che …