If we suppose that we start with an instance of k$k$-SAT, and try converting the problem to an instance of (k+m)$(k+m)$-SAT, where there are $(k+m)$ literals per clause, can we garaunteeguarantee a reduction in the total amount of clauses?
I realized after posting that we can't gauranteeguarantee that the number of clauses can be reduced. However, I wonder if we have $n$ clauses, could we get something like $n/k + O(1)$ clauses by some "reduction" technique?
If so, how much can we garaunteeguarantee the total number of clauses can be reduced by? ForFor instance, if we start with k$k$-SAT with $n_k$ clauses, what is the smallest garaunteedguaranteed $n_{k+m}$, the new amount of clauses, that will result if we convert this instance to (k+m)$(k+m)$-SAT?
More importantly, how do we carry out this conversion?