To address the given problem, we start by interpreting the key terms and constraints involved. The problem describes a set of cities with airlines (edges) connecting them, which can be considered as a graph where cities are vertices and airlines are edges. We need to determine whether there exists a subset of exactly 2022 cities (vertices) for which it is impossible to two-color the connecting airlines (edges) without forming a monochromatic triangle.

### Step 1: Understanding the Ramsey Number Concept

The core mathematical concept involved here is related to Ramsey theory. In Ramsey theory, the Ramsey number $R(k, l)$ is defined as the minimum number of vertices $n$ such that any graph of $n$ vertices, when its edges are colored with two colors, will contain a monochromatic complete subgraph with $k$ vertices in one color or $l$ vertices in the other color.

Specifically, $R(3, 3)$ is the smallest number of vertices required to ensure any two-coloring of the edges results in a monochromatic triangle. From known results:

$$R(3, 3) = 6.$$

This implies that any set of 6 vertices in a graph will always contain a monochromatic triangle if the edges are colored with two colors.

### Step 2: Applying the Ramsey Number to the Given Problem

We are asked if there is always a subset of exactly 2022 cities such that it is impossible to color the airlines between them in two colors without a monochromatic triangle.

To investigate this, consider a subset of 5 cities. The Ramsey number tells us it is possible to color the airlines between 5 cities in two colors without forming a monochromatic triangle. Thus, for any subset of cities with fewer than 6 vertices, a two-coloring exists where no monochromatic triangles are formed.

### Conclusion

Given that $R(3, 3) = 6$, there can be a complete graph or any subset with more than 5 vertices forming an "unlucky" set due to the two-coloring condition. However, for exactly 2022 cities, it is not guaranteed they will always form an unlucky set because the presence of a monochromatic triangle is contingent upon the exact structure and coloring.

Hence, the answer to whether there is always an unlucky set containing exactly 2022 cities is:

$$\boxed{\text{No}}$$

There might exist colorings and configurations that avoid a monochromatic triangle among exactly 2022 cities, hence not every arrangement of 2022 cities results in an unlucky set.