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To solve the given problem using the contradiction method, we are asked to prove that for all \( n = 2, 3, 4, \dots \), there exists an undirected graph \( G = (V, E) \) where \( |V| = n \), which:

1. **Has** \( \frac{(n-1)(n-2)}{2} \) edges.
2. **Is not connected**.

### Contradiction method steps:

1. **Assume the negation of the statement**:
   Assume that for all \( n \geq 2 \), any undirected graph \( G = (V, E) \) with \( |V| = n \) and \( \frac{(n-1)(n-2)}{2} \) edges **is connected**.

2. **Properties of a connected graph**:
   A graph is connected if there is a path between every pair of vertices. For a graph with \( n \) vertices, the maximum number of edges is \( \frac{n(n-1)}{2} \), which corresponds to a complete graph \( K_n \).

3. **Edge count comparison**:
   The given number of edges \( \frac{(n-1)(n-2)}{2} \) can be simplified and compared to the total possible number of edges in a complete graph:
   \[
   \frac{(n-1)(n-2)}{2} = \frac{n^2 - 3n + 2}{2}
   \]
   which is less than \( \frac{n(n-1)}{2} \), the total number of edges in a complete graph. This implies that we have fewer edges than a complete graph, suggesting the possibility of disconnected vertices.

4. **Construction of a disconnected graph**:
   Consider a graph where one vertex \( v_0 \) is isolated (has no edges). The remaining \( n-1 \) vertices form a complete subgraph \( K_{n-1} \). This subgraph will have exactly \( \frac{(n-1)(n-2)}{2} \) edges. This construction satisfies both conditions:
   - The graph has \( \frac{(n-1)(n-2)}{2} \) edges.
   - The graph is not connected because \( v_0 \) is isolated.

5. **Contradiction**:
   The assumption that a graph with \( \frac{(n-1)(n-2)}{2} \) edges is always connected leads to a contradiction, as we have demonstrated a construction where such a graph is not connected.

### Conclusion:
Thus, by contradiction, we have proven that for all \( n \geq 2 \), there exists an undirected graph \( G = (V, E) \) with \( |V| = n \) and \( \frac{(n-1)(n-2)}{2} \) edges that is not connected.

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Do you need further clarification or more details on any part of the proof?

Here are five related questions to explore further:

1. How does the edge count of a complete graph \( K_n \) differ from the given formula?
2. Can you prove the same statement using an alternative method such as induction?
3. What is the significance of the condition \( n \geq 2 \)?
4. How can the concept of graph connectivity be extended to weighted graphs?
5. What is the minimum number of edges required to ensure a graph is connected for any \( n \)?

**Tip:** When working with graph theory, always consider constructing specific cases of graphs to gain better insights into general properties.

Prove: for all n = 2, 3, 4, ... there exists an undirected graph G = (V, E) with |V| = n, which has (n-1)(n-2)/2 edges and is not connected.

In this proof, we explore the existence of an undirected graph with n vertices, (n-1)(n-2)/2 edges, that is not connected. Using concepts from graph theory, including edge counting and graph connectivity, this problem illustrates how isolated vertices affect graph properties. Suitable for undergraduate math students.

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