Atiyah conjecture for free groups. It has been proved by Peter Linnel using some operator-algebraic technique, but the statement seems to me to be ultimately combinatorial.
For example, as an important special case there is the analytical 0-divisors conjecture: Let T be a self-adjoint element of the complex group ring of a free group of $l^2$-norm at most 1. Consider the sequence $t_n$ of complex numbers: $t_n$ is the coefficient of the neutral element of the element $(1-T)^n$ of the group ring (so this is a combinatorial thing.)
One of the formulations of the analytical 0-divisors conjecture is the following theorem.
Theorem (P. Linnel): If T is not 0 then the limit of the sequence $t_n$ is 0.
Similarly for many other groups for which Atiyah conjecture is known. For example, the proof for elementary amenable group (again Linnel) uses deep K-theory, but admittedly it might be that this deep K-theory is proven using combinatorics.