Let us consider a set of communities, denoted as vertices in a graph, where each edge between a pair of communities is labeled with one of the following modes of transportation: bus, train, or airplane. The problem imposes the following conditions:

1. All three modes of transportation (bus, train, and airplane) are used.
2. No community is serviced by all three modes.
3. No three communities are linked pairwise by the same mode of transportation.

We are tasked with finding the largest possible number of communities, $n$, in this county satisfying these conditions.

### Step-by-step Analysis:

1. Graph Representation:
Each community is a vertex, and each connection (bus, train, airplane) between two communities is an edge labeled with a transportation mode. The objective is to find the maximum number of vertices.

2. Condition Application:
- Since at least one connection must use each mode, each transportation mode must appear on some edge at least once.
- No vertex can have all three different connections due to the restriction regarding any community being unable to be serviced by all three transportation modes.
- No three vertices form a complete subgraph (a triangle) with all edges having the same label.

3. Exploring Possibilities:
- If we consider 3 communities (vertices), we can assign each pair a unique mode of transportation. That satisfies all conditions: no vertex will have all three modes, and we won't have a triangle of the same transportation mode.
- Trying to add a fourth community is where the challenge arises. If we add another community and attempt to connect it with the existing three while using three distinct labels, it becomes complex given the restrictions.

4. Complete Solution:
- We can construct a scenario with 4 communities where each mode appears on one pair and none meets the forbidden conditions. Assign specific modes to avoid forming a triangle with the same mode or having any vertex connected by all three modes. This can be achieved by careful choice of modes:
- Label connections (1,2) and (3,4) with mode 1, (1,3) and (2,4) with mode 2, and (1,4) and (2,3) with mode 3.
- If we try to extend beyond 4 communities, adhering to all conditions will force overlaps where communities either receive all three modes or form a complete same-mode triangle.

Thus, by confirming all conditions are met with 4 communities and observing the difficulty in maintaining them with more, we conclude that the maximum number of communities that satisfy all given conditions is:

$$\boxed{4}$$