  Fink, M\'esz\'aros and St.Dizier showed that the Schubert polynomial
$\mathfrak{S}_w(x)$ is zero-one if and only if $w$ avoids twelve permutation
patterns. In this paper, we prove that the Grothendieck polynomial
$\mathfrak{G}_w(x)$ is zero-one, i.e., with coefficients either 0 or $\pm$1, if
and only if $w$ avoids six patterns. As applications, we show that zero-one
homogeneous Grothendieck polynomials are Lorentzian, partially confirming three
conjectures of Huh, Matherne, M\'esz\'aros and St.Dizier. Moreover, we verify
several conjectures on the support and coefficients of Grothendieck polynomials
posed by M\'{e}sz\'{a}ros, Setiabrata and St.Dizier for the case of zero-one
Grothendieck polynomials.
