  Let $f:\mathbb{N}\to \mathbb{D}$ be a multiplicative function. Under the
merely necessary assumption that $f$ is non-pretentious (in the sense of
Granville and Soundararajan), we show that for any pair of distinct integer
shifts $h_1,h_2$ the two-point correlation
  $$\frac{1}{x}\sum_{n\leq x}{f(n+h_1)\overline{f}(n+h_2)}$$
  tends to $0$ along a set of $x\in\mathbb{N}$ of full upper logarithmic
density. We also show that the same result holds for the $k$-point correlations
$$\frac{1}{x}\sum_{n\leq x}{f(n+h_1)\cdots f(n+h_k)}$$ if $k$ is odd and $f$ is
a real-valued non-pretentious function. Previously, the vanishing of
correlations was known only under stronger non-pretentiousness hypotheses on
$f$ by the works of Tao, and Tao and the third author. We derive several
applications, including:
  (i) A classification of $\pm 1$-valued completely multiplicative functions
that omit a length four sign pattern, solving a 1974 conjecture of R.H. Hudson.
  (ii) A proof that a class of "Liouville-like" functions satisfies the
unweighted Elliott conjecture of all orders, solving a problem of de la Rue.
  (iii) Constructing examples of multiplicative $f:\mathbb{N}\to \{-1,0,1\}$
with a given (unique) Furstenberg system, answering a question of Lema\'nczyk.
  (iv) A density version of the Erd\H{o}s discrepancy theorem of Tao.
