Problem 1726

1. Problem Restatement and Setup:
We are given a school with $n$ subjects, $2^n$ students, and $2^{n-1}$ rooms. Each student is interested in a unique subset of the $n$ subjects. We need to prove that the students can be arranged in a circle such that:
- Each pair of roommates stands next to each other.
- For any pair of students standing next to each other who are not roommates, one student is interested in all the subjects the other student is interested in, plus exactly one more subject.

2. Graph Construction:
Construct a graph $G$ where each vertex represents a student. Two vertices are adjacent if one student is interested in all the subjects the other student is interested in, plus exactly one more subject. This graph $G$ is an $n$-dimensional hypercube.

3. Pairing and Alternating Cycle:
If two students are roommates, pair the corresponding vertices. We need to show that for any pairing $P$ of the vertices of $G$, there exists a sequence of $2^n$ distinct vertices that alternates between edges of $P$ and edges of $G$. This sequence is called an *alternating cycle*.

4. Induction Base Case:
For $n = 1$, there are 2 students and 1 room. The graph $G$ is a 1-dimensional hypercube (a line segment). The students can trivially be arranged in a circle (a loop) satisfying the conditions.

5. Inductive Step:
Assume the statement is true for $n-1$. We need to prove it for $n$.

6. **Choosing a Subject $X$:**
Choose a subject $X$. Consider two cases:
- Case 1: No two vertices differing in the $X$-coordinate are paired under $P$.
- Define subgraphs $G_0$ and $G_1$ of $G$, where $G_0$ contains vertices not interested in $X$ and $G_1$ contains vertices interested in $X$.
- Partition $P$ into $P_0$ and $P_1$, where $P_0$ pairs vertices within $G_0$ and $P_1$ pairs vertices within $G_1$.
- By the inductive hypothesis, find alternating cycles $C_0$ in $G_0$ and $C_1$ in $G_1$.
- Combine $C_0$ and $C_1$ into a single alternating cycle by collapsing the $X$-coordinate and using the resulting pairings $S_0'$ and $S_1'$ to form a cycle $K$ in the collapsed graph $G'$.
- Transform $S_1$ to $T_1$ by replacing edges corresponding to $K$ and form the final alternating cycle $C$ by combining $P$, $S_0$, and $T_1$.

- Case 2: There exist vertices differing in the $X$-coordinate paired under $P$.
- These edges are called *spanning edges*. There must be an even number of spanning edges.
- Separate vertices into sets $V_0$ and $V_1$ corresponding to $G_0$ and $G_1$.
- Erase spanning edges and replace them within $V_0$. Use the inductive hypothesis to find an alternating cycle $C_0$ in $V_0$.
- Restore the original pairing $P$ and break $C_0$ into fragments $F_1, F_2, \ldots, F_m$.
- Match these fragments with edges $l_1, l_2, \ldots, l_m$ in $V_1$ and use the inductive hypothesis to find an alternating cycle $C_1$ in $V_1$.
- Combine $C_0$ and $C_1$ by restoring the original pairing $P$ and filling in the missing edges with spanning edges and fragments, forming the final alternating cycle $C$.

7. Conclusion:
By induction, the statement holds for all $n$. Thus, the students can be arranged in a circle satisfying the given conditions.