The indivisibility of the homogeneous Kn-free graphs

Abstract We will prove that for each n ≥ 3 the homogeneous Kn-free graph Hn is indivisible. That means that for every partition of Hn into two classes R and B there is an isomorphic copy of Hn in R or in B. This extends a result of Komjath and Rodl [Graphs Combin., 2 (1986), 55–60] who have shown that H3 is indivisible.