This question is inspired from


(i) Theorems like the "universal friend theorem": If every two vertices in a connected graph $G$ share a unique common neighbor, then there is a vertex connected to all the others in $G$.


and (ii) Results like: If the subgraph spanned by every $k$ vertices in $G$ is $2$-colorable, then $\chi(G)=O(n^{O(1/k)})$.


Unfortunately I don't know many results similar in flavor to the above, therefore the question. What are some important theorems/principles/methods in graph theory that help us determine global parameters of the graph from local data? (I am being intentionally vague about what I mean by "local", examples could vary from data on subgraphs spanned by few vertices, to data on subgraphs spanned by vertices at small distance from a base vertex)



Local-global approach to graph theory
There is a very nice survey [Local-global phenomena in graphs](http://www.cs.huji.ac.il/~nati/PAPERS/local_global.pdf) by N. Linial


