Unavoidable induced subgraphs in large graphs with no homogeneous sets

A homogeneous set of an n-vertex graph is a set X of vertices (2 <= vertical bar X vertical bar <= n - 1) such that every vertex not in X is either complete or anticomplete to X. A graph is called prime if it has no homogeneous set. A chain of length t is a sequence of t+1 vertices such that for every vertex in the sequence except the first one, its immediate predecessor is its unique neighbor or its unique non-neighbor among all of its predecessors. We prove that for all n, there exists N such that every prime graph with at least N vertices contains one of the following graphs or their complements as an induced subgraph: (1) the graph obtained from K-1,K-n by subdividing every edge once, (2) the line graph of K-2,K-n, (3) the line graph of the graph in (1), (4) the half-graph of height n, (5) a prime graph induced by a chain of length n, (6) two particular graphs obtained from the half-graph of height n by making one side a clique and adding one vertex. (C) 2016 Elsevier Inc. All rights reserved.