The notion of $k$ -truss has been introduced a decade ago in social network analysis and security for community detection, as a form of cohesive subgraphs less stringent than a clique (set of pairwise linked nodes), and more selective than a k-core (induced subgraph with minimum degree $k$ ). A $k$ -truss is an inclusion-maximal subgraph $H$ in which each edge belongs to at least $k-2$ triangles inside $H$ . The truss decomposition establishes, for each edge $e$ , the maximum $k$ for which $e$ belongs to a $k$ -truss. Analogously to the largest clique and to the maximum $k$ -core, the strongest community for $k$ -truss is the max-truss, which corresponds to the $k$ -truss having the maximum $k$ . Even though the computation of truss decomposition and of the max-truss takes polynomial time, on a large scale, it suffers from handling a potentially cubic number of wedges. In this paper, we provide a new algorithm FMT, which advances the state of the art on different sides: lower execution time, lower memory usage, and no need for expensive hardware. We compare FMT experimentally with the most recent state-of-the-art algorithms on a set of large real-world and synthetic networks with over a billion edges. The massive improvement allows FMT to compute the max-truss of networks of tens of billions of edges on a single standard server machine.