TITLE: Meeting of people.
QUESTION [7 upvotes]: In a group of k people, some are acquainted with each other and some are not. There are two rooms for dinner. Every person chooses to stay in that room, in which he has an even number of acquaintances. Prove that the number of different ways that people can be divided in these rooms is always a power of 2.
I've tried to switch it in a graph problem, considering every person as a point and connecting each two points with an edge if they are acquainted. Then we know that the number of odd- degree points is even. But I don't know how to proceed. Any help would be appreciated. Thanks in advance.

REPLY [4 votes]: This is an answer for a question from this comment. 

Do you perhaps know any other combinatorial problem which could be solved with an abstract or linear algebra?

I know several applications of an abstract or linear algebra in combinatorics. 
1) David Ellis, “Algebraic Methods in Combinatorics”.  The paper starts as follows: 

In the last ﬁfty years, algebraic methods have been used with striking success in combinatorics. This course looks at some of the most important of these methods, and some of the most beautiful results obtained using them. We will also explore connections with combinatorial geometry, probability theory and theoretical computer science.

2) I guess  a book “Thirty-three Miniatures: Mathematical and Algorithmic Applications of Linear Algebra” by Jiřì Matoušek also contains related miniatures. 
3) There is a paper “Combinatorial Nullstellensatz” by Noga Alon. Its abstract says: 

We present a general algebraic technique and discuss some of its numerous applications in Combinatorial Number Theory, in Graph Theory and in Combinatorics. These applications include results in additive number theory and in the study of graph coloring problems. Many of these are known results, to which we present united proofs, and some results are new.

Googing for this paper, I also found a few related papers:
– Evan Chen, “Combinatorial Nullstellensatz”.
– Andrew Drucker, “Notes on the Combinatorial Nullstellensatz”.
– Brad R. Jones, “Combinatorial Nullstellensatz”. 
– Zoltán Lóránt Nagy, “Applications of the Combinatorial Nullstellensatz”, Ph.D. thesis.
– Pete L. Clark, “The Combinatorial Nullstellensätze Revisited”.
4) Abstract and linear algebra are used for construction of Steiner systems and more general block designs.
5) Now concerning the papers of people working in this direction, which I know personally (because they are from Ukraine). There is a (small?) book “Methods of Linear Algebra in Combinatorics” by I. V. Protasov and O. M. Khromulyak. Unfortunately, it is in Ukraininan and I don’t have it. I have only an unnamed (and, probably, overlapping with it) Protasov’s book of sketches in combinatorics, contaning a 17-pages section  “Linear Algebra in Combinatorics” with problems and theorems.  In patrticular, it contains Fisher-Bose(?) theorem, Chaudhuri(?)–Wilson theorems, Kahn and Kalai solution of Borsuk’s problem (based on a theorem by Frankl and  Wilson), and  Bollobás’(?) theorem. Unfortunately, the book  is also in Ukrainian and I don’t have a permission to share it (it was sent by Ihor Protasov to Taras Banakh, who sent it to me). Since the first link for further reading at this Wikipedia page is broken, I say that  Oleg Pikhurko’s course notes “Algebraic Methods in Combinatorics” are here.
6) At last, recently I proved an algebraic lemma (see here, it order to show that some integer linear programming problem has an optimal solution with a relatively simply form. This result was used in order to provide an algorithm to solve a combinatorial optimization problem (see a paper “Computing Optimal Tangles Faster” by Oksana Firman, Philipp Kindermann, Alexander Wolff, Johannes Zink and me). The problem input data complexity is estimated  by natural numbers $n$ and $l$, and in the considered case $l$ is extremely large. The proposed algorithm is so complicated that it is still unpublished, but I expect its computation complexity to be about $\exp(O(n^7\log n)\log l)$. One can say that this is a very slow algorithm, but I stated with an algorithm of  complexity like $\exp(\exp(\exp O(n\log n)))\log l$.