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Our goal is to investigate a close relative of the independent transversalproblem in the class of infinite $K_n$-free graphs: we show that for anyinfinite $K_n$-free graph $G=(V,E)$ and $m\in \mathbb N$ there is a minimal$r=r(G,m)$ such that for any balanced $r$-colouring of the vertices of $G$ onecan find an independent set which meets at least $m$ colour classes in a set ofsize $|V|$. Answering a conjecture of S. Thomass\'e, we express the exact valueof $r(H_n,m)$ (using Ramsey-numbers for finite digraphs), where $H_n$ isHenson's countable universal homogeneous $K_n$-free graph. In turn, we deduce anew partition property of $H_n$ regarding balanced embeddings of bipartitegraphs: for any finite bipartite $G$ with bipartition $A,B$, if the vertices of$H_n$ are partitioned into two infinite classes then there is an induced copyof $G$ in $H_n$ such that the images of $A$ and $B$ are contained in differentclasses.

Balanced independent sets in graphs omitting large cliques