Let $M$ be a compact manifold, and let $M_1,\ldots, M_k$ (k>2) be embedded submanifolds. Suppose that $p\in\cap_{i=1}^k M_k$ and that for any subset $S$ of $\{1,\ldots, k\}$ and any $j\notin S$ that $\cap_{i\in S}M_i$ intersects $M_j$ transversally at $p$.

I believe that in this case the fact that $\cap_{i=1}^k M_k$ is nonempty is stable (still true after homotoping each $M_i$ a little bit). Does anyone have a reference for this fact?