= Solution
Under a uniform compressive <pressure> $p$, each normal <stress> is $-p$. The isotropic <Hooke's law> gives each longitudinal <strain> as
$$
\varepsilon_x=\varepsilon_y=\varepsilon_z
=\frac{-p-\sigma(-p-p)}E=-\frac{p(1-2\sigma)}E.
$$
Thus $\Delta V/V=-3p(1-2\sigma)/E$. By the definition of <bulk modulus>, $B=-p/(\Delta V/V)$, so the <isotropic elastic modulus relation> is
$$
\boxed{B=\frac E{3(1-2\sigma)},\qquad E=3B(1-2\sigma).}
$$
Positive finite <Young's modulus> and <bulk modulus> require
$$
\boxed{\sigma<\frac12.}
$$
The incompressible limit is $\sigma\to1/2$ with $B\to\infty$ at finite $E$. If positivity of the <shear modulus> $G_s=E/[2(1+\sigma)]$ is also imposed, the full stable isotropic range is $-1<\sigma<1/2$. Negative <Poisson's ratio> is not ruled out; the displayed bulk-modulus relation alone supplies the upper bound.
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