We show that a randomly chosen

$3$-CNF formula over $n$ variables with clauses-to-variables

ratio at least $4.4898$ is asymptotically almost surely unsatisfiable.

The previous best such bound,

due to Dubois in 1999, was $4.506$.

The first such bound, independently

discovered by many groups of researchers since 1983,

was $5.19$. Several decreasing values between

$5.19$ and $4.506$ were published in the years between.

The probabilistic techniques we use for the proof are, we believe, of independent interest.