A novel weighting scheme for random $k$-SAT

Abstract: Consider a random $k$-CNF formula $F_{k}(n, rn)$ with $n$ variables and $rn$ clauses. For every truth assignment $\sigma\in {0, 1}^{n}$ and every clause $c=\ell_{1}\vee\cdots\vee\ell_{k}$, let $d=d(\sigma, c)$ be the number of satisfied literal occurrences in $c$ under $\sigma$. For fixed $\beta>-1$ and $\lambda>0$, we take $\omega(\sigma, c)=0$, if $d=0$; $\omega(\sigma, c)=\lambda(1+\beta)$, if $d=1$ and $\omega(\sigma, c)=\lambda^{d}$, if $d>1$. Applying the above weighting scheme, we get that if $F_{k}(n, rn)$ is unsatisfiable with probability tending to one as $n\rightarrow\infty$, then $r\geq2.83, 8.09, 18.91, 40.81, 84.87$ for $k=3, 4, 5, 6$ and $7,$ respectively.