Among the 2020 students who came for the internship, some know each other (acquaintances are mutual; new acquaintances do not appear during the task; we assume that the person does not know himself). It is known that for any group of students, there will be one among them who knows no more than 583 people in the selected group. What is the largest number of pairs of acquaintances that can be among the students?

This problem can be tackled using graph theory, where each student is represented as a vertex, and an acquaintance between two students is represented as an edge between two vertices. The goal is to determine the largest number of edges (acquaintances) possible, given the constraint that in any subset of students, there is always at least one student who knows no more than 583 other people.

### Rephrasing the Problem:
- We have a group of \( n = 2020 \) students, and the relationship between them can be described using a graph with 2020 vertices.
- Each vertex has a degree representing the number of people a student knows (i.e., the number of edges connected to that vertex).
- The key condition is that **in any subset of students, there is always one student who knows no more than 583 others**.

### Analyzing the Condition:
This condition implies that the **minimum degree** of any vertex is bounded. In other words, for any subset of students, the student who knows the fewest people (the minimum degree) must have a degree of at most 583.

We need to find the largest number of edges in this graph. The number of edges in a graph is limited by the degrees of the vertices.

### Step-by-Step Approach:

1. **Maximizing the Number of Edges:**
   In an unrestricted complete graph with \( n \) vertices, the maximum number of edges is given by:
   \[
   \text{Maximum edges} = \frac{n(n-1)}{2}
   \]
   For \( n = 2020 \), this would be:
   \[
   \frac{2020(2019)}{2} = 2039190
   \]
   However, the given condition restricts how densely connected the graph can be.

2. **Considering the Degree Bound:**
   The constraint states that in any subset, there is at least one student who knows no more than 583 other students. This implies that the maximum degree of any vertex (student) must be at most 583. If any subset had all students knowing more than 583 others, the condition would be violated.

3. **Applying the Degree Bound:**
   Since each vertex (student) can know at most 583 others, the graph's maximum degree is constrained to 583. The number of edges in the graph, therefore, cannot exceed:
   \[
   \text{Maximum edges} \leq \frac{n \times \Delta}{2}
   \]
   where \( \Delta \) is the maximum degree, and \( n = 2020 \).

   Substituting \( \Delta = 583 \) and \( n = 2020 \):
   \[
   \text{Maximum edges} \leq \frac{2020 \times 583}{2} = 589430
   \]

Thus, the largest number of pairs of acquaintances that can be among the 2020 students, given the constraint, is **589,430**.

### Summary:
The largest number of pairs of acquaintances (edges in the graph) that can exist among the 2020 students is **589,430**, considering the constraint that in any subset of students, there is always one who knows no more than 583 others.

Do you want further clarifications or details? Here are some related questions to deepen your understanding:

1. How does graph theory help in solving problems related to social networks?
2. What is the significance of the degree of a vertex in graph theory?
3. How would the result change if the number of students or the degree bound were different?
4. Can we apply similar principles to find the maximum number of edges in a bipartite graph?
5. How can the concept of a graph’s degree sequence be used in practical applications like networking?

**Tip:** In graph theory, the degree of a vertex is a crucial measure that can help understand the connectivity and structure of the network.

Explore how to maximize the number of acquaintance pairs among 2020 students using graph theory. This problem involves understanding vertex degrees and calculating the maximum number of edges with constraints. Learn how to apply degree bounds and combinatorics to find that the maximum number of pairs is 589,430.

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