The notion of a simplicial complex seems to be pretty standard. We can count the number of faces of various dimensions to get the face numbers of the complex. for example, a complex may have 7 vertices and 11 edges. The face vector of a complex lists its face numbers.

The chromatic number of a simplicial complex is the minimum number of colors needed to color its vertices such that no two vertices in a face are the same color. This must always be at least as large as the number of vertices in the face with the most vertices. If this is exactly the number of vertices in the largest face, we call the complex balanced. The face vectors of balanced complexes were characterized by Frankl, Furedi, and Kalai in the mid-1980s.

A simplicial complex is called a flag complex if all minimal non-faces are two element sets. Flag complexes are the same thing as clique complexes of graphs. We show that the face vector of every flag complex is also the face vector of some balanced complex. That is, face vectors of flag complexes satisfy the characterization for face vectors of balanced complexes. This was first conjectured independently by Kalai and Eckhoff around 1990. The reverse does not hold: for example, it is easy to construct a balanced complex with (1, 4, 5, 1) as its face vector, but there is no flag complex with that face vector.

The proof is constructive, and only about two pages long. Given a flag complex, we construct a balanced complex with the same face numbers in two consecutive dimensions. The structure of the above result of Frankl, Furedi, and Kalai shows that this is enough.