A homogeneous set of an n-vertex graph is a set X of vertices (2≤∣X∣≤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 K1,n by subdividing every
edge once, (2) the line graph of K2,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.Comment: 13 pages, 3 figure