We propose models of combinatorial algorithms for the Boolean Matrix Multiplication (BMM), and prove lower bounds on computing BMM in these models. First, we give a relatively relaxed combinatorial model which is an extension of the model by Angluin (1976), and we prove that the time required by any algorithm for the BMM is at least $Omega(n^3 / 2^O(\sqrt \log n ))$. Subsequently, we propose a more general model capable of simulating the "Four Russians Algorithm". We prove a lower bound of $Omega(n^7/3 / 2^O(\sqrt \log n ))$ for the BMM under this model. We use a special class of graphs, called $(r,t)$-graphs, originally discovered by Rusza and Szemeredi (1978), along with randomization, to construct matrices that are hard instances for our combinatorial models.

Joint work with Debarati Das and Mike Saks.