Divide* the unit square $I^{2}=[0,1]\times[0,1]$ into polygonal regions, each having the property that the distance between any two of its points is less than $\frac{1}{30}$.
Question: Must there be a polygon $P$ within $I^{2}$ surrounded by at least six adjacent polygons- that is, touching $P$ in at least a point? If yes, how to prove it?
*The polygons in the division may not be convex and the division may have "gaps" in the sense that the boundary between two adjacent polygons may be a disconnected polygonal line. The picture below illustrates gaps (white polygons) between the two grey polygons $P$ and $P'$.
$\hskip2.25in$ 
For example, the division pictured below has gaps.
$\hskip2in$ 
I think the answer may be yes from some drawings. And I do not think it matters if there are gaps. Indeed, these gaps will be produced by polygons in the original division. We can merge them with adjacent polygons in order to produce larger polygons and yield a new division of $I^{2}$ such that any two adjacent polygonal regions have either a point or a connected polygonal line as boundary between them. Since this process only decreases the number of adjacent polygons a polygon has then if the answer is affirmative for this new division, then it must also be affirmative for the original division. But that is as far as I have been able to go towards a solution.
Best Answer
Yes! That is, assuming no gaps.
Form a connected path along the boundaries of regions from one side of the square to the opposite side. Now, consider the boundaries of all the regions touching the path. You get two bands of regions. Repeat to get three adjacent bands. If there were at most 5 adjacent regions, each region in the middle band must either touch only one region in the lower band or only one region in the upper band. Moreover, each region in the lower and upper bands can touch at most two regions in the middle band. So, we get a leapfrog pattern like the one shown. Then, you have a regions in the upper and lower bands touching 3 regions in the middle band, two in their own band and at least one more outside any of the three bands.