In a class with $23$ students, each pair of students have watched a movie together. Let the set of movies watched by a student be his [i]movie collection[/i]. If every student has watched every movie at most once, at least how many different movie collections can these students have?
1. **Understanding the Problem:**
   We have a complete graph \( K_{23} \) where each vertex represents a student and each edge represents a movie watched by the pair of students connected by that edge. We need to determine the minimum number of different movie collections among the students.

2. **Graph Theory Application:**
   In \( K_{23} \), each vertex has a degree of \( 22 \) because each student has watched a movie with every other student. We need to color the edges such that no two edges incident to the same vertex share the same color. This is equivalent to finding an edge coloring of the graph.

3. **Edge Coloring Theorem:**
   According to Vizing's Theorem, the chromatic index (the minimum number of colors needed to color the edges of a graph such that no two edges incident to the same vertex share the same color) of a simple graph \( G \) is either \( \Delta(G) \) or \( \Delta(G) + 1 \), where \( \Delta(G) \) is the maximum degree of the graph.

4. **Applying Vizing's Theorem:**
   For \( K_{23} \), the maximum degree \( \Delta(G) \) is \( 22 \). Therefore, the chromatic index is either \( 22 \) or \( 23 \). Since \( K_{23} \) is a complete graph, it falls into the category of graphs where the chromatic index is \( \Delta(G) + 1 \). Thus, the chromatic index is \( 23 \).

5. **Interpreting the Result:**
   This means we need at least \( 23 \) different colors to color the edges of \( K_{23} \). Each color represents a different movie. Therefore, there are at least \( 23 \) different movies.

6. **Movie Collections:**
   Each student has watched \( 22 \) movies, and each movie collection is unique to each student. Since there are \( 23 \) students, and each student has a unique collection of \( 22 \) movies out of the \( 23 \) different movies, the minimum number of different movie collections is \( 23 \).

\[
\boxed{23}
\]