Around the log-rank conjecture

The log-rank conjecture states that the communication complexity of a boolean matrix A is bounded by a polynomial in the log of the rank of A . Equivalently, it says that the chromatic number of a graph is bounded quasi-polynomially in the rank of its adjacency matrix. This old conjecture is well known among computer scientists and mathematicians, but despite extensive work it is still wide open. We survey results relating to the log-rank conjecture, describing the current state of affairs and collecting related questions. Most of the results we discuss are well known, but some points of view are new. One of our hopes is to paint a path to the log-rank conjecture that is made of a series of smaller questions, which might be more feasible to tackle.