TITLE: Islands and bridges
QUESTION [8 upvotes]: I stole this question from another site :)
The archipelago of Wonderland consists of 131072 islands. For any two islands, there is a bridge that connects them. There are 17 managing companies that service at least one of these bridges each. The servicing personnel have decided to go on strike and therefore bring out of service some of the bridges but in such a way that with the remaining bridges, all islands can be accessed from any other island, either directly or through some other island. It has therefore been agreed that only some of the 17 companies will allow their personnel to go on strike. What is the maximum number of companies that can safely allow their personnel to go on strike, given the above condition? 
Really challenging but I have no clue what to do!

REPLY [4 votes]: This answer is to build on the answer of Jens to prove that exactly $8$ companies can go on strike, and no more. I will formalize Jens's construction that shows at most $8$ companies can go on strike, and show that it is always possible for at least $8$ companies to go on strike.
Let the companies be $1,2,\dots, 17$. Consider a set $S$ of $\binom{17}{9}$ islands, and label them each with a unique subset of $\{1,2,\dots,17\}$ of size $9$. Note that $\binom{17}{9}<2^{17}$ so we have at least that many islands. Now between any two islands in $S$, the subsets they were assigned share at least one number $x$. Assign company $x$ to the bridge between those islands. For each bridge between an island in $S$ and an island not in $S$, assign a company at random among the companies in the subset assigned to the island in $S$. For each bridge between two islands not in $S$, assign any company at random. Now in this configuration, each island in $S$ can only be reached by bridges of the $9$ companies in the subset assigned to it, and there is such an island corresponding to every set of $9$ companies. So if any $9$ companies are selected to go on strike, the island corresponding to those companies will be disconnected. This shows that there is a configuration that allows at most $8$ companies to go on strike.
Now suppose, for contradiction, that there is some configuration that only allows $7$ companies to go on strike, and no more. Then if some set $X$ of $8$ companies go on strike, an island (or group of islands) will be disconnected from the others. Consider what happens if the other $17-8=9$ companies go on strike (call this set of companies $Y$). I claim that the bridges of the companies in $X$ connect all the islands, which is a contradiction to the assumption that only $7$ companies can go on strike. Consider two islands $A$ and $B$. If the bridge between the two islands is operated by a company in $X$, then $A$ and $B$ are connected by that bridge. So assume that the bridge between $A$ and $B$ is not operated by one of the companies in $X$. Now when the $X$ companies go on strike, the bridge between $A$ and $B$ is still operating (it's run by one of the companies in $Y$), but the islands are disconnected, so there must be a third island $C$ that cannot be reached by $A$ and $B$ via bridges run by companies in $Y$. In particular, the bridges between $A$ and $C$ and between $B$ and $C$ are run by companies in $X$, and there is path between $A$ and $B$ via bridges run by companies in $X$. Note that this is essentially an argument that for any graph $G$, either $G$ or its complement is connected. And this shows that in any configuration, at least $8$ companies can go on strike.
I would also like to point out that this argument generalizes to the scenario when you have $k$ companies and at least $\binom{k}{\left \lceil \frac{k}{2} \right \rceil}$ islands, then exactly $\left \lfloor \frac{k}{2} \right \rfloor$ companies can go on strike, and no more. It seems that the situation in which there are fewer than $\binom{k}{\left \lceil \frac{k}{2} \right \rceil}$ islands would be more complex, but it would be interesting to know if there are any results in this direction.