  We show that there can be no finite list of conditional independence
relations which can be used to deduce all conditional independence implications
among Gaussian random variables. To do this, we construct, for each $n> 3$ a
family of $n$ conditional independence statements on $n$ random variables which
together imply that $X_1 \ind X_2$, and such that no subset have this same
implication. The proof relies on binomial primary decomposition.
