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? 



Normal intersections of submanifolds
The matter being local, we can restrict to a nbd $U$ of $p$ and think that $M\_i$ is the zero set of some local submersion $g\_i:U\to\mathbb{R}^{n\_i}$. If I'm not wrong your transversality assumption then translates into the surjectivity of the differential at $p$ of the map $g:=(g\_1,\dots,g\_k):U\to\mathbb{R}^m$ (here $m:={n\_1+\dots+n\_k}$). So $0\in\mathbb{R}^m$ is a regular value for $g$, which implies your thesis. Note that the compactness assumption on $M$ plays no role.


