Linear Time Algorithm for Computing a Small Biclique in Graphs without Long Induced Paths

The biclique problem asks, given a graph G and a parameter k, whether G has a complete bipartite subgraph of k vertices in each part (a biclique of order k). Fixed-parameter tractability of this problem is a longstanding open question in parameterized complexity that received a lot of attention from the community. In this paper we consider a restricted version of this problem by introducing an additional parameter s and assuming that G does not have induced (i.e. chordless) paths of length s. We prove that under this parameterization the problem becomes fixed-parameter linear. The main tool in our proof is a Ramsey-type theorem stating that a graph with a long (not necessarily induced) path contains either a long induced path or a large biclique.