Gaussian conditional independence relations have no finite complete characterization

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.