For 2-CNF formulas the maximum number of accepted weight-t assignments is q^{n-t-r}(q+1)^r, and for t=n-k the general problem is equivalent to the Turán problem.
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On Extremal Properties of k-CNF: Capturing Threshold Functions
For 2-CNF formulas the maximum number of accepted weight-t assignments is q^{n-t-r}(q+1)^r, and for t=n-k the general problem is equivalent to the Turán problem.