We study new partition properties of infinite K_n-free graphs. First, we investigate the number bpi(G,m) introduced by A. Aranda et al. (denoted there by r(G,m)) : the minimal r so that for any partition of G into r classes of equal size, there exists an independent set which meets at least m classes in size |G|. In the case of Henson's countable universal K_n-free graph H_n, we express bpi(H_nm) by well-known Ramsey-numbers for finite digraphs. In particular we answer a conjecture of Thomassé (2000) by showing that indeed bpi(H_n,2)=2 for all n⩾3. Furthermore, we bound and in some cases determine bpi(G,m) for certain geometric graphs, including shift graphs, unit distance graphs and orthogonality graphs. Second, we prove a new universal partition property for H_n. Given a finite bipartite graph G on classes A,B, we show that for any balanced 2-colouring of H_n there is an induced copy of G in H_n so that the images of A and B are monochromatic of distinct colours. We end our paper with further open problems.